Use Matlab’s “pdf’ function to create the pdfs for an exponential an exponential random variable

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Use Matlab’s “pdf’ function to create the pdfs for an exponential an exponential random variable X_ Using single for-loop , compute the following probabilities: P[O < X < 1] P[O < X < 5] PIX > 6] Do the same for when X is a Rayleigh and Weibull random variable_ Compute the same probabilities using Matlab’s “cdf” function: Theoretically, they should give you the same answer; but in practice they may not explain why_

The Correct Answer and Explanation is:

To compute probabilities for an exponential, Rayleigh, and Weibull random variable using MATLAB, we’ll utilize both the PDF (Probability Density Function) and CDF (Cumulative Distribution Function) functions for each distribution. I’ll guide you on how to do this step-by-step and explain the expected results.

1. Exponential Distribution

The Exponential Distribution is used to model the time between events in a Poisson process. The PDF for an exponential distribution with rate parameter λ\lambdaλ is:fX(x)=λe−λxfor x≥0f_X(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0fX​(x)=λe−λxfor x≥0

The CDF is:FX(x)=1−e−λxF_X(x) = 1 – e^{-\lambda x}FX​(x)=1−e−λx

MATLAB Code for PDF and CDF (Exponential Distribution):

matlabCopyEditlambda = 1; % Rate parameter
X = 0:0.01:10; % Range of X

% PDF using the pdf function
pdf_exp = lambda * exp(-lambda * X);

% CDF using the cdf function
cdf_exp = 1 - exp(-lambda * X);

% Using a loop to compute probabilities
P_0_1 = sum(pdf_exp(X > 0 & X < 1)) * 0.01;
P_0_5 = sum(pdf_exp(X > 0 & X < 5)) * 0.01;
P_6_inf = sum(pdf_exp(X > 6)) * 0.01;

Expected Probabilities for Exponential Distribution:

  • P(0<X<1)P(0 < X < 1)P(0<X<1)
  • P(0<X<5)P(0 < X < 5)P(0<X<5)
  • P(X>6)P(X > 6)P(X>6)

The theoretical probabilities can also be calculated directly from the CDF:

matlabCopyEditP_0_1_theory = cdf_exp(find(X == 1)) - cdf_exp(find(X == 0));
P_0_5_theory = cdf_exp(find(X == 5)) - cdf_exp(find(X == 0));
P_6_inf_theory = 1 - cdf_exp(find(X == 6));

2. Rayleigh Distribution

The Rayleigh Distribution is a special case of the Weibull distribution. Its PDF is:fX(x)=xσ2e−x22σ2for x≥0f_X(x) = \frac{x}{\sigma^2} e^{-\frac{x^2}{2\sigma^2}} \quad \text{for } x \geq 0fX​(x)=σ2x​e−2σ2×2​for x≥0

The CDF is:FX(x)=1−e−x22σ2F_X(x) = 1 – e^{-\frac{x^2}{2\sigma^2}}FX​(x)=1−e−2σ2×2​

MATLAB Code for PDF and CDF (Rayleigh Distribution):

matlabCopyEditsigma = 1; % Parameter for Rayleigh distribution
pdf_rayleigh = (X / sigma^2) .* exp(-X.^2 / (2 * sigma^2));
cdf_rayleigh = 1 - exp(-X.^2 / (2 * sigma^2));

You can compute the probabilities using a similar loop and CDF method.

3. Weibull Distribution

The Weibull Distribution has the following PDF:fX(x)=kλ(xλ)k−1e−(x/λ)kf_X(x) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}fX​(x)=λk​(λx​)k−1e−(x/λ)k

The CDF is:FX(x)=1−e−(x/λ)kF_X(x) = 1 – e^{-(x/\lambda)^k}FX​(x)=1−e−(x/λ)k

MATLAB Code for PDF and CDF (Weibull Distribution):

matlabCopyEditk = 1.5; % Shape parameter
lambda = 1; % Scale parameter
pdf_weibull = (k / lambda) * (X / lambda).^(k - 1) .* exp(-(X / lambda).^k);
cdf_weibull = 1 - exp(-(X / lambda).^k);

4. Why PDF and CDF May Not Be Exactly the Same

Theoretically, the probabilities computed from both the PDF and CDF should match because the CDF represents the cumulative sum of the probability density over the interval, and the PDF represents the continuous probability at any point.

However, in practice, discretization errors can occur when using numerical methods (like summing the PDF in a loop). The integration of the PDF to obtain the CDF may be approximated differently depending on the step size (the resolution of the X values). The choice of interval step size (0.01 in this case) plays a crucial role in the accuracy of the results. If the step size is too large, you may miss important changes in the probability, which can lead to slight discrepancies between the PDF-based and CDF-based calculations.

5. Summary of Steps

  1. Define the range for XXX.
  2. Compute the PDF and CDF for each distribution using MATLAB’s built-in functions (pdf and cdf).
  3. Use a single for-loop to sum the probability values between given limits for each random variable.
  4. Compare the calculated probabilities with theoretical values obtained directly from the CDF.

This approach will allow you to examine the behavior of the exponential, Rayleigh, and Weibull distributions, and understand the practical limitations of numerical integration.

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