Probability distributions in MATLAB

Introduction

Probability distributions help us understand how likely different outcomes are in a random process.

To find the chance of rolling a certain number on a dice.

To model the heights of people in a group.

To predict the number of emails you get in an hour.

To analyze test scores and their spread.

To simulate random events in games or experiments.

Syntax

MATLAB

pd = makedist('DistributionName', 'Parameter1', value1, 'Parameter2', value2, ...);
prob = pdf(pd, x);
cumProb = cdf(pd, x);
randomSamples = random(pd, n, 1);

makedist creates a probability distribution object.

pdf gives the probability density or mass at a point.

Examples

This creates a normal distribution with mean 0 and standard deviation 1, then finds the probability density at 0.

MATLAB

pd = makedist('Normal', 'mu', 0, 'sigma', 1);
prob = pdf(pd, 0);

This creates a binomial distribution for 10 trials with 0.5 success chance, then finds the cumulative probability up to 5 successes.

MATLAB

pd = makedist('Binomial', 'N', 10, 'p', 0.5);
cumProb = cdf(pd, 5);

This creates a Poisson distribution with average 3, then generates 5 random samples from it.

MATLAB

pd = makedist('Poisson', 'lambda', 3);
samples = random(pd, 5, 1);

Sample Program

This program creates a normal distribution with mean 5 and standard deviation 2. It calculates the probability density and cumulative probability at the value 7, then prints the results.

MATLAB

pd = makedist('Normal', 'mu', 5, 'sigma', 2);
value = 7;
prob_density = pdf(pd, value);
cum_probability = cdf(pd, value);
fprintf('Probability density at %.1f is %.4f\n', value, prob_density);
fprintf('Cumulative probability up to %.1f is %.4f\n', value, cum_probability);

Important Notes

Use makedist to create different distributions like Normal, Binomial, Poisson, etc.

pdf is for probability density (continuous) or mass (discrete) at a point.

cdf gives the chance of a value being less than or equal to a point.

Summary

Probability distributions describe how likely outcomes are.

Use makedist to create distributions in MATLAB.

pdf and cdf help find probabilities at or below points.

Practice

1. Which MATLAB function is used to create a probability distribution object?

easy

A. rand

B. makedist

C. histogram

D. plot

Solution

1. Step 1: Understand the purpose of each function

   makedist creates a probability distribution object, while rand generates random numbers, histogram plots data distribution, and plot creates graphs.

2. Step 2: Identify the correct function for creating distributions

   Only makedist is designed to create probability distribution objects in MATLAB.

3. Final Answer:

   makedist -> Option B

4. Quick Check:

   Function to create distribution = makedist [OK]

Hint: Use makedist to create distributions in MATLAB [OK]

Common Mistakes:

• Confusing makedist with rand
• Using histogram to create distributions
• Trying to use plot for distributions

2. Which of the following is the correct syntax to create a normal distribution with mean 0 and standard deviation 1 in MATLAB?

easy

A. pd = makedist('normal', 0, 1);

B. pd = makedist('Normal', 'mean', 0, 'std', 1);

C. pd = makedist('Normal', 0, 1);

D. pd = makedist('Normal', 'mu', 0, 'sigma', 1);

Solution

1. Step 1: Recall the correct parameter names for normal distribution

   In MATLAB, the normal distribution parameters are specified as 'mu' for mean and 'sigma' for standard deviation.

2. Step 2: Check the syntax for makedist

   The correct syntax uses name-value pairs: makedist('Normal', 'mu', 0, 'sigma', 1). Other options use incorrect parameter names or formats.

3. Final Answer:

   pd = makedist('Normal', 'mu', 0, 'sigma', 1); -> Option D

4. Quick Check:

   Normal dist params = 'mu' and 'sigma' [OK]

Hint: Use 'mu' and 'sigma' as parameter names for normal dist [OK]

Common Mistakes:

• Using lowercase 'normal' instead of 'Normal'
• Using 'mean' and 'std' instead of 'mu' and 'sigma'
• Passing parameters without name-value pairs

3. What is the output of the following MATLAB code?

pd = makedist('Exponential', 'mu', 2);
p = pdf(pd, 1);
disp(round(p, 3));

medium

A. 0.303

B. 0.368

C. 0.500

D. 0.607

Solution

1. Step 1: Understand the exponential distribution PDF formula

   The PDF for exponential with mean mu=2 is f(x) = (1/2)*exp(-x/2). For x=1, f(1) = 0.5 * exp(-0.5) ≈ 0.5 * 0.6065 = 0.3033.

2. Step 2: Compute and round the value

   p ≈ 0.3033, round(p, 3) = 0.303.

3. Final Answer:

   0.303 -> Option A

4. Quick Check:

   (1/mu)*exp(-x/mu) at x=1, mu=2 ≈ 0.303 [OK]

Hint: Recall pdf = (1/mu)*exp(-x/mu) for exponential [OK]

Common Mistakes:

• Ignoring the (1/mu) factor
• Confusing PDF with CDF
• Rounding errors causing mismatch

4. Identify the error in this MATLAB code snippet:

pd = makedist('Binomial', 'n', 10, 'p', 0.5);
prob = cdf(pd, 11);

medium

A. The probability p must be an integer

B. makedist does not support 'Binomial' distribution

C. The cdf input value 11 exceeds the number of trials n

D. The variable name 'prob' is reserved

Solution

1. Step 1: Understand binomial distribution parameters

   Binomial distribution with n=10 means possible outcomes are from 0 to 10.

2. Step 2: Check the cdf input value

   Calling cdf(pd, 11) asks for probability of outcomes ≤ 11, but 11 is outside the valid range (max 10).

3. Step 3: Identify the error

   Input to cdf must be within 0 to n. Using 11 is invalid and causes an error or unexpected result.

4. Final Answer:

   The cdf input value 11 exceeds the number of trials n -> Option C

5. Quick Check:

   cdf input must be ≤ n for binomial [OK]

Hint: cdf input must be within distribution support range [OK]

Common Mistakes:

• Using cdf input outside valid range
• Thinking p must be integer
• Assuming makedist lacks binomial support

5. You want to calculate the probability that a normally distributed variable with mean 5 and standard deviation 2 is less than 7 in MATLAB using a probability distribution object. Which code correctly computes this?

hard

A. pd = makedist('Normal', 'mu', 5, 'sigma', 2); p = cdf(pd, 7);

B. p = normcdf(7, 5, 2);

C. pd = makedist('Normal', 5, 2); p = pdf(pd, 7);

D. p = cdf('Normal', 7, 5, 2);

Solution

1. Step 1: Understand how to create a normal distribution object

   Use makedist('Normal', 'mu', 5, 'sigma', 2) to create the distribution with mean 5 and std 2.

2. Step 2: Use cdf to find probability less than 7

   Calling cdf(pd, 7) returns the cumulative probability up to 7.

3. Step 3: Check other options

   p = normcdf(7, 5, 2); uses normcdf which is valid but not using makedist. pd = makedist('Normal', 5, 2); p = pdf(pd, 7); incorrectly uses pdf instead of cdf. p = cdf('Normal', 7, 5, 2); has wrong syntax for cdf.

4. Final Answer:

   pd = makedist('Normal', 'mu', 5, 'sigma', 2); p = cdf(pd, 7); -> Option A

5. Quick Check:

   Use makedist + cdf for cumulative probability [OK]

Hint: Use cdf with makedist object for cumulative probability [OK]

Common Mistakes:

• Using pdf instead of cdf
• Wrong syntax for cdf
• Ignoring distribution parameters