Probabilistics Distribution.

BERNOULLI DISTRIBUTION

1. Definition

The Bernoulli distribution is the most fundamental discrete probability distribution. It describes an experiment or trial that only has two possible outcomes, usually labeled as "success" and "failure".

Mathematically:

1. Success: $P(X = 1) = p,$

2. Failure: $P(X = 0) = 1-p$

A single experiment of this type is known as a Bernoulli trial. This distribution therefore models the outcome of a single trial or event, not a sequence of them.

2. How and When Is It Used?

It is used to model any situation in which a single event has a binary outcome. It is the basis for more complex distributions such as the Binomial and Geometric.

Some application examples are:

• Coin toss: The result is heads (success) or tails (failure).

• Quality inspection: A manufactured part is conforming (success) or defective (failure).

• Exam result: A student passes (success) or fails (failure).

• Customer decision: A customer buys a product (success) or does not buy it (failure).

3. Algebraic Form

To formalize it, a random variable X is defined that can take two values:

• X=1 if the outcome is "success".

• X=0 if the outcome is "failure".

The Probability Mass Function (pmf) of a Bernoulli random variable is:

$$f(x;p)=P(X=x)=p^x(1−p)^{1−x} \quad \text{for}\ x∈\{0,1\}$$

• If x=1 (success): $P(X=1)=p^1(1−p)^{1−1}=p(1−p)^0=p$

• If x=0 (failure): $P(X=0)=p^0(1−p)^{1−0}=1(1−p)^1=1−p$

Main statistical measures:

• Expected Value (Mean): $E[X]=p$

• Variance: $Var(X)=p(1−p)$

4. In Excel

Excel does not have a dedicated function like BERNOULLI.DIST. However, a Bernoulli trial is a special case of the Binomial distribution where the number of trials is 1 (n=1). Therefore, the syntax is: BINOM.DIST(successes; trials; prob_success; cumulative)

To simulate a Bernoulli trial:

• successes: The outcome to evaluate (1 for success, 0 for failure).

• trials: Always 1.

• prob_success: Probability of success, p.

• cumulative: Always FALSE (to get the exact value probability).

Example: To calculate the probability that a part is defective (p=0.05), the result is "success" if X=1. =BINOM.DIST(1; 1; 0.05; FALSE) returns 0.05.

5. How to Use It in Simulation (AnyLogic)

In system simulation, the Bernoulli trial is crucial for modeling probabilistic decision points. The implementation does not require sampling from a complex distribution, but simply:

Logic:

1. Generate a random number R from a Uniform(0, 1) distribution.

2. If R≤p, the result is "success".

3. If R>p, the result is "failure".

In AnyLogic, this is greatly simplified with the built-in function:

randomTrue(p)

This function returns the boolean value true (success) with probability p, and false (failure) with probability 1-p. It is the direct implementation of a Bernoulli trial.

Example use: In a quality inspection model, an entity arrives at a SelectOutput block. In the condition of that block, you can write randomTrue(0.98). This would mean...

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BINOMIAL DISTRIBUTION

1. Definition

The Binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of n independent Bernoulli trials.

For a random variable to follow a Binomial distribution, the experiment must meet these fundamental conditions:

1. The experiment consists of a sequence of n identical and fixed trials.

2. Each trial only has two possible outcomes: "success" or "failure".

3. The probability of success, denoted by p, is constant in each trial. The probability of failure is 1−p.

4. The trials are statistically independent; the outcome of one does not affect another.

This distribution is defined by two parameters: n (number of trials) and p (probability of success).

2. How and When Is It Used?

It is used to model how many times an event of interest occurs in a fixed number of attempts. It's one of the most widely used distributions in quality control, genetics, finance, etc.

Examples:

• Quality control: Probability of finding exactly k defective items in a lot of n units drawn from a production line.

• Medicine: Probability that k patients out of n respond positively to a treatment.

• Marketing: Probability that k customers out of n who received a promotion make a purchase.

• Games of chance: Probability of getting exactly k heads when tossing a coin n times.

3. Algebraic Form

Let X be the random variable representing the number of successes in n trials. The Probability Mass Function (pmf) of the Binomial distribution is:

$$f(k;n,p)=P(X=k)={n \choose k}p^k(1−p)^{n−k}$$

where:

• k is the number of successes, taking values {0,1,2,...,n}

• ${n \choose k} = \frac{n!}{k!(n−k)!}$ is the binomial coefficient, the number of ways to have k successes in n trials.

• $p^k$ is the probability of k successes.

• $(1−p)^{n−k}$ is the probability of (n−k) failures.

Main statistical measures:

• Expected Value (Mean): $E[X]=np$

• Variance: $Var(X)=np(1−p)$

4. In Excel

Excel provides a direct function for Binomial distribution: BINOM.DIST (or DISTR.BINOM in Spanish).

Syntax: BINOM.DIST(successes; trials; prob_success; cumulative)

• successes: Value of k for which the probability is calculated.

• trials: Total number of trials, n.

• prob_success: Probability of success, p.

• cumulative:

  • FALSE: Calculates probability of exactly k successes (P(X=k)).

  • TRUE: Calculates probability of up to k successes (P(X≤k)), i.e., the cumulative distribution function (CDF).

Example: In a manufacturing process, if the probability of a defective part is 0.02 (p=0.02) and a sample of 50 parts (n=50) is taken, what's the probability of finding...

5. How to Use It in Simulation (AnyLogic)

In simulation, the Binomial distribution is used to determine the aggregate outcome of a series of binary events without simulating each of the n trials individually.

In AnyLogic, you can sample from a Binomial distribution directly:

binomial(n, p)

This function performs n Bernoulli trials with probability of success p and returns the total number of successes.

Example: A ship arrives at a port with 200 containers (n=200). If the probability that a container is selected for inspection is 0.15 (p=0.15), instead of simulating 200 randomTrue(0.15) events, you can use:

int containers_to_inspect = binomial(200, 0.15);

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POISSON DISTRIBUTION

1. Definition

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.

Unlike the Binomial, which counts successes in a fixed number of trials (n), Poisson counts the number of occurrences in a continuum (such as time or area).

• λ (lambda): The average rate of occurrence of events in the specified interval.

2. How and When Is It Used?

One of the most important distributions in operations research and engineering, especially in queueing theory. Used to model events considered "rare" or that occur randomly in time or space.

Examples:

• Number of phone calls arriving at a switchboard in one hour.

• Number of defects per square meter of steel sheet.

• Number of customers arriving at a bank or checkout per minute.

• Number of machine breakdowns in a day.

Most common use in simulation: modeling arrival processes.

3. Algebraic Form

Let X be the random variable representing the number of occurrences in an interval. The Probability Mass Function (pmf) for Poisson is:

$$f(k;\lambda)=P(X=k)=\frac{\lambda^k e^{−\lambda}}{k!}$$

where:

• k is the number of occurrences, taking non-negative integer values {0,1,2,...}

• λ is the average rate per interval.

• e is the base of natural logarithms (about 2.71828...)

Unique property: The expected value and variance are both equal to λ.

• Expected Value (Mean): $E[X]=\lambda$

• Variance: $Var(X)=\lambda$

4. In Excel

Excel function: POISSON.DIST (or POISSON in Spanish).

Syntax: POISSON.DIST(x; mean; cumulative)

• x: Number of events, k.

• mean: Mean rate, λ.

• cumulative:

  • FALSE: Probability of exactly k events (P(X=k)).

  • TRUE: Probability of up to k events (P(X≤k)), i.e., CDF.

Example: If customers arrive at a checkout at a mean rate of 15 per hour (λ=15), what is the probability that exactly 10 arrive in an hour (k=10)? =POISSON.DIST(10; 15; FALSE)

5. How to Use It in Simulation (AnyLogic)

In discrete-event simulation, Poisson is rarely used directly for arrivals, because models do not advance in steps of "one arrival". Instead, the time between successive events is modeled.

There is a fundamental relationship between Poisson and Exponential:

If the number of events in a time interval follows a Poisson distribution with mean rate λ, then the time between successive events follows an Exponential distribution with mean 1/λ.

To model a Poisson arrival process in AnyLogic, use the Exponential distribution in the Source block:

exponential(1.0 / lambda)

Direct use of poisson() in AnyLogic: AnyLogic has a function poisson(lambda) for generating an integer count, not for interarrival times.

Examples:

• For arrivals: A workshop receives jobs according to a Poisson process with mean 2 per hour (λ=2). In Source, set interarrival time: exponential(0.5)

• For counts: A truck arrives daily; the number of packages it unloads follows a Poisson distribution with mean 30. Use poisson(30) for package count.

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GEOMETRIC DISTRIBUTION

1. Definition

The Geometric distribution is a discrete probability distribution that models the number of independent Bernoulli trials until the first success is observed.

Like Binomial, it's based on Bernoulli trials, so:

1. Each trial has only two outcomes: "success" or "failure".

2. The probability of success, p, is constant in each trial.

3. Trials are statistically independent.

Key difference: Geometric does not have a fixed number of trials (n). The experiment continues until the first success.

2. How and When Is It Used?

It is used to model "waiting for the first event" situations in discrete, independent trials. Useful in quality control, reliability, and telecommunications.

Examples:

• Manufacturing: How many items must be inspected to find the first defective one?

• Telecommunications: Number of times a data packet must be retransmitted until received without errors.

• Sales: Number of prospects a salesperson must contact before making the first sale.

• Games of chance: Number of dice rolls until the first "6" appears.

3. Algebraic Form

There are two common variants. The most common models the number of failures before the first success.

Let X be the random variable: number of failures before the first success. The Probability Mass Function (pmf) is:

$$f(k;p)=P(X=k)=(1−p)^k\,p\quad \text{for}\ k=0,1,2,\ldots$$

• k: number of failures.

• p: probability of success.

• (1−p): probability of failure.

Main statistical measures:

• Expected Value (Mean): $E[X]=\frac{1-p}{p}$ (expected failures before first success)

• Variance: $Var(X)=\frac{1-p}{p^2}$

(Note: The other variant defines the variable as the total number of trials, $k'$, to achieve first success.)

4. In Excel

Newer Excel versions include GEOM.DIST, but the most compatible universal function is NEGBINOM.DIST (or DISTR.NEGBINOM in Spanish), as the Geometric distribution is a special case of the Negative Binomial.

Syntax: NEGBINOM.DIST(num_failures; num_successes; prob_success; cumulative)

• num_failures: k (number of failures)

• num_successes: Always 1

• prob_success: Probability of success, p

• cumulative: Always FALSE

Example: A basketball player has an 80% chance (p=0.8) of making a free throw. What is the probability of missing exactly 2 times (k=2) before making the first success? =NEGBINOM.DIST(2; 1; 0.8; FALSE)

5. How to Use It in Simulation (AnyLogic)

In simulation, the Geometric distribution is useful for modeling the number of attempts until a random action succeeds.

In AnyLogic, sample directly with:

geometric(p)

This simulates Bernoulli trials with probability of success p and returns the total number of trials until the first success.

Example: If a software check has a 95% chance (p=0.95) of passing each time, simulate how many attempts are needed:

int numAttempts = geometric(0.95);

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DISCRETE UNIFORM DISTRIBUTION

1. Definition

The Discrete Uniform Distribution describes an experiment with a finite number of possible outcomes, each equally likely. The mathematical formalization of "pure chance".

Defined by the number of possible outcomes, n.

2. How and When Is It Used?

Used for situations where every discrete result has the same chance of occurring.

• Games of chance: Rolling a fair die, where each face (1-6) is equally likely.

• Sampling: Drawing a card at random from a well-shuffled deck.

• Computing: Generating a random integer within a specific range.

3. Algebraic Form

Let X be a random variable that can take n distinct values {x₁, x₂, ..., xₙ}. The pmf is: $P(X=x_i) = \frac{1}{n}$ for each i=1,2,...,n.

If the results are consecutive integers from a to b, n = b−a+1.

• Expected Value (Mean): $E[X]=\frac{a+b}{2}$

• Variance: $Var(X)=\frac{(b−a+1)^2−1}{12}$

4. In Excel

No specific function for probability (it's just 1/n), but to generate a random sample:

RANDBETWEEN(bottom, top) or ALEATORIO.ENTRE(inferior; superior) (Spanish)

Returns a random integer between bottom and top (inclusive), equally likely.

Example: To simulate a roll of a 6-sided die: =RANDBETWEEN(1, 6)

5. How to Use It in Simulation (AnyLogic)

In AnyLogic, to select a random integer result from a range where all are equally likely:

uniform_discrete(min, max)

Returns a random integer between min and max, inclusive.

Example: If an assembly process must randomly pick one of four screw types (numbered 1–4): uniform_discrete(1, 4)

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CONTINUOUS UNIFORM DISTRIBUTION

1. Definition

The Continuous Uniform Distribution describes a random variable that can take any value in an interval [a,b] with equal probability.

Defined by two parameters: a (min value) and b (max value).

2. How and When Is It Used?

Widely used in simulation, especially when the process range (min and max) is known but there's insufficient info for another distribution.

• Operation time modeling: A manual task time varies between 5 and 10 minutes, no value is more likely than another.

• Initial estimates: When no historical data, experts usually give a "pessimistic-optimistic" range, which fits a uniform distribution.

3. Algebraic Form

Let X be a continuous random variable in [a, b].

• Probability Density Function (pdf): $f(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b; 0 \text{ otherwise}$

• Cumulative Distribution Function (CDF): $F(x) = \frac{x-a}{b-a} \text{ for } a \leq x \leq b$

• Expected Value (Mean): $E[X]=\frac{a+b}{2}$

• Variance: $Var(X)=\frac{(b-a)^2}{12}$

4. In Excel

To generate a sample: use RAND(), which returns a decimal between 0 and 1, scaled to [a, b]:

Formula: =a + (b-a) * RAND()

Example: Generate a service time uniformly between 5 and 10 minutes: =5 + (10-5) * RAND()

5. How to Use It in Simulation (AnyLogic)

AnyLogic provides a direct function:

uniform(min, max)

Returns a floating-point number uniformly distributed between min and max.

Example: For a shoe factory, the operation time for cut Type 2 is uniformly between 45 and 60 seconds: uniform(45, 60)

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EXPONENTIAL DISTRIBUTION

1. Definition

The Exponential distribution is a continuous probability distribution that describes the time between two successive events in a Poisson process.

Defined by:

• λ (lambda): Average event rate per unit time.

A fundamental property: memorylessness (the probability of an event in the future does not depend on how much time has already elapsed).

2. How and When Is It Used?

The classic distribution for modeling time between random, independent events.

• Queueing theory: Most common for modeling times between arrivals and service times (e.g., M/M/1 queue).

• Reliability engineering: Time until failure of components with no aging (constant failure rate).

• Physics: Radioactive decay.

3. Algebraic Form

Let T be the continuous random variable for time between events.

• pdf: $f(t)=\lambda e^{−\lambda t},\quad t \geq 0$

• CDF: $F(t)=P(T\leq t)=1−e^{−\lambda t}$

• Expected Value: $E[T]=\frac{1}{\lambda}$

• Variance: $Var(T)=\left(\frac{1}{\lambda}\right)^2$

4. In Excel

Excel function: EXPON.DIST (or DISTR.EXPON in Spanish).

Syntax: EXPON.DIST(x; lambda; cumulative)

• x: Time value t.

• lambda: Rate λ.

• cumulative:

  • FALSE: Value of the density function.

  • TRUE: Cumulative probability (CDF).

Example: Customers arrive at a bank at rate 20 per hour (λ=20). What's the probability the next customer arrives within 3 minutes (0.05 hours)? =EXPON.DIST(0.05; 20; TRUE)

5. How to Use It in Simulation (AnyLogic)

Most used distribution for modeling random entity arrivals (Poisson process) in AnyLogic Source blocks.

In AnyLogic, the function is:

exponential(mean)

Note: exponential() in AnyLogic takes the mean inter-event time, not λ.

Example:

• Arrivals: To model a Poisson arrivals process with mean 10 customers/hour (λ=10), the mean time between arrivals is 1/10=0.1 hours. In Source, use: exponential(0.1)

• Service Time: If service time is exponential with mean 12 minutes: exponential(12)

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NORMAL DISTRIBUTION

1. Definition

The Normal distribution, also known as the Gaussian distribution or bell curve, is the most important continuous probability distribution in statistics and engineering.

It is bell-shaped, symmetric about its center. Defined by:

• μ (mu): The mean, expected value and center (highest point).

• σ (sigma): The standard deviation, measuring dispersion.

2. How and When Is It Used?

Widely applicable because many natural phenomena and engineering processes follow its pattern, due to the Central Limit Theorem.

Used to model:

• Physical Characteristics: Height, weight, part dimensions.

• Measurement Errors: Random experimental errors.

• Financial Processes: Asset returns.

• Process Times: Operation times clustered around an average.

3. Algebraic Form

Let X be a continuous random variable.

• pdf: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2},\quad -\infty < x < \infty$

• CDF: Integral of the density (no simple closed form; use tables/software).

• Expected Value: $E[X] = \mu$

• Variance: $Var(X) = \sigma^2$

For probabilities: standardize X to Z (standard normal): $Z = \frac{X-\mu}{\sigma}$

4. In Excel

Direct functions: NORM.DIST (DISTR.NORM in Spanish).

Syntax: NORM.DIST(x; mean; std_dev; cumulative)

• x: Random variable value.

• mean: μ.

• std_dev: σ.

• cumulative:

  • TRUE: CDF (P(X≤x))

  • FALSE: Density function

Example: In a filling process, bottle volume ~ N(500,5) ml. Probability a bottle contains 498 ml or less: =NORM.DIST(498; 500; 5; TRUE)

5. How to Use It in Simulation (AnyLogic)

Excellent for modeling process times or product attributes with a clear target value and symmetrical random variation.

In AnyLogic:

normal(mean, stdDev)

• mean: μ.

• stdDev: σ.

Example: For shoe factory, assembly time (Station 2) is normally distributed with mean 3 min, std dev 1 min: normal(3, 1)

Important Warning: The Normal distribution can theoretically generate negative values. For physical variables like time, use:

max(0, normal(mean, stdDev))

Ensures negative values are replaced with zero.

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TRIANGULAR DISTRIBUTION

1. Definition

The Triangular distribution is a continuous probability distribution defined by three parameters that determine its triangular shape:

• a: Minimum possible value

• b: Maximum possible value

• c: Most likely value (mode)

Unlike the infinite Normal, the Triangular is defined on finite [a, b], making it practical for bounded phenomena.

2. How and When Is It Used?

Ideal when you lack large amounts of historical data but have expert opinion (minimum, most likely, maximum).

Common applications:

• Project Management: Basis of PERT technique for estimating project activity durations.

• Service/Process Times: Estimate times for manual operations.

• Risk Analysis: Model financial/business variables with subjective estimates.

3. Algebraic Form

Let X be a continuous random variable.

• pdf:

  $$f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)}, & a \leq x \leq c \\ \frac{2(b-x)}{(b-a)(b-c)}, & c < x \leq b \end{cases}$$
• Expected Value: $E[X]=\frac{a+b+c}{3}$

• Variance: $Var(X)=\frac{a^2+b^2+c^2-ab-ac-bc}{18}$

4. In Excel

No built-in TRIANG.DIST. To generate a sample, use the inverse transform method with a Uniform(0,1) random number.

Logic:

1. Generate a random number R with RAND()

2. Calculate threshold $F_c = (c-a)/(b-a)$

3. If R < F_c, use formula for ascending ramp.

4. If R ≥ F_c, use formula for descending ramp.

Due to its complexity, this is usually implemented in Excel via macros (VBA) or simulation add-ins, not a direct cell formula.

5. How to Use It in Simulation (AnyLogic)

The Triangular distribution is widely used in AnyLogic for process times due to the ease of estimating its parameters.

In AnyLogic:

triangular(min, mode, max)

• min: Minimum, a.

• mode: Most likely, c.

• max: Maximum, b.

Example: In Borshchev's AnyLogic book, the time for a process is modeled with triangular(0.5, 1, 1.5) seconds (min 0.5, max 1.5, mode 1).

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ERLANG DISTRIBUTION

1. Definition

The Erlang distribution is a continuous probability distribution describing the total time until k successive events occur in a Poisson process.

It can be understood as the sum (convolution) of k identical, independent exponential variables with the same mean rate.

Defined by:

• k: Shape parameter, positive integer (number of exponential events).

• λ (lambda): Mean event rate of the underlying Poisson process.

As k increases, the distribution becomes more symmetric, approaching the Normal (Central Limit Theorem).

2. How and When Is It Used?

Used to model waiting times or task durations that can be seen as the sum of several sequential, independent phases.

Applications:

• Queueing Theory: Service times with several phases (e.g., diagnosis, repair, test).

• Supply Chain: Delivery time through k distribution centers or stages.

• Reliability Engineering: Time until failure of a system with k sequential backup components.

3. Algebraic Form

Let T be the total time to the k-th event.

• pdf: $f(t)=\frac{\lambda^k t^{k−1}e^{−\lambda t}}{(k−1)!},\quad t≥0,\,k=1,2,...$

• Expected Value: $E[T]=\frac{k}{\lambda}$

• Variance: $Var(T)=\frac{k}{\lambda^2}$

4. In Excel

No direct ERLANG.DIST, but Erlang is a special case of the Gamma distribution (shape parameter is integer).

Syntax: GAMMA.DIST(x; alpha; beta; cumulative)

• x: Total time, t.

• alpha: Shape parameter, k.

• beta: Scale, which is mean time per phase, 1/λ.

• cumulative: TRUE for CDF, FALSE for pdf.

Example: A process has 4 stages (k=4), event rate 2/min (λ=2). Probability process takes 3 min or less: =GAMMA.DIST(3; 4; 0.5; TRUE)

5. How to Use It in Simulation (AnyLogic)

Useful for process times less variable than the Exponential (whose peak is always at zero), but not as symmetric as Normal.

In AnyLogic:

erlang(k, mean)

• k: Shape parameter, number of stages.

• mean: Total mean time for the process (k/λ).

Example: Assembly process has two sequential phases (k=2), total mean time is 30 min: erlang(2, 30) (each phase has mean 15 min).

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WEIBULL DISTRIBUTION

1. Definition

The Weibull distribution is a highly versatile continuous probability distribution. Its main advantage is its ability to model different failure rate behaviors.

Defined by:

• α (alpha): Shape parameter (key to the distribution's character)

• β (beta): Scale parameter (characteristic life), point where 63.2% of failures have occurred.

2. How and When Is It Used?

Classic in reliability engineering, lifetime analysis, and warranty management.

• α < 1: Failure rate decreases over time ("infant mortality", early product failures)

• α = 1: Constant failure rate. Weibull reduces to the Exponential.

• α > 1: Failure rate increases over time (wear-out failures in mechanics).

3. Algebraic Form

Let T be the time to failure.

• pdf: $f(t) = \frac{\alpha}{\beta} \left( \frac{t}{\beta} \right)^{\alpha-1} e^{-(t/\beta)^\alpha},\quad t \geq 0$

• Expected Value: $E[T]=\beta\cdot\Gamma(1+1/\alpha)$

• Variance: $Var(T) = \beta^2 [\Gamma(1+2/\alpha) - (\Gamma(1+1/\alpha))^2]$

4. In Excel

Direct function: WEIBULL.DIST (or DISTR.WEIBULL in Spanish).

Syntax: WEIBULL.DIST(x; alpha; beta; cumulative)

• x: Time value, t.

• alpha: Shape parameter, α.

• beta: Scale parameter, β.

• cumulative: TRUE for CDF, FALSE for pdf.

Example: A component has α=2 (wear), β=8000 hours. Probability it fails before 7000 hours: =WEIBULL.DIST(7000; 2; 8000; TRUE)

5. How to Use It in Simulation (AnyLogic)

Use for process durations or failure times with non-constant failure rates.

In AnyLogic:

weibull(shape, scale)

• shape: α

• scale: β

Example: For a motor lifetime with α=2.5, β=15000 hours: weibull(2.5, 15000)

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LOGNORMAL DISTRIBUTION

1. Definition

The Lognormal distribution is a continuous probability distribution for a variable whose natural logarithm is normally distributed. Since the log of a negative number is undefined, it only applies to positive variables.

It is right-skewed, ideal for phenomena with many small values and rare large extremes.

2. How and When Is It Used?

Applies to variables thought to be the product of many small, independent factors (Central Limit Theorem applied multiplicatively).

Examples:

• Finance: Stock prices, asset values.

• Engineering/Medicine: Repair times, incubation periods, chemical concentrations, particle sizes.

3. Algebraic Form

• pdf: $f(x) = \frac{1}{x \sigma_{ln} \sqrt{2\pi}} e^{-\frac{(\ln(x) - \mu_{ln})^2}{2 \sigma_{ln}^2}},\quad x > 0$

  • Important: $\mu_{ln}$ and $\sigma_{ln}$ are the mean and standard deviation of the natural logarithm of the variable, not the variable itself.

• Expected Value: $E[X] = e^{\mu_{ln} + \sigma_{ln}^2 / 2}$

• Variance: $Var(X) = (e^{\sigma_{ln}^2} - 1) e^{2\mu_{ln} + \sigma_{ln}^2}$

4. In Excel

Direct function: LOGNORM.DIST (or DISTR.LOG.NORM in Spanish).

Syntax: LOGNORM.DIST(x; mean; std_dev; cumulative)

• x: Variable value.

• mean: Mean of ln(X), i.e., $\mu_{ln}$.

• std_dev: Std dev of ln(X), i.e., $\sigma_{ln}$.

• cumulative: TRUE for CDF, FALSE for pdf.

5. How to Use It in Simulation (AnyLogic)

Great for process times that can't be negative and are skewed: most times are short but occasionally there's a long process.

In AnyLogic:

lognormal(mean, stdDev)

• mean: The mean of the lognormal distribution itself, E[X].

• stdDev: The std dev of the lognormal distribution.

Note: AnyLogic simplifies usage. No need to calculate $\mu_{ln}$ and $\sigma_{ln}$ of the log; just provide the observed mean and std dev.

Example: If machine diagnostics time has observed mean 45 min and std dev 20 min (and is known to be right-skewed): lognormal(45, 20)