determine the mean value of 〈𝑥^2〉 using the distribution 𝜋(𝑥) = 𝑒^(−|𝑥|) through random sampling in matlab

To determine the mean value of 〈𝑥^2〉 using the distribution 𝜋(𝑥) = 𝑒^(−|𝑥|) through random sampling in Matlab, you can follow these steps:

1. Define the distribution 𝜋(𝑥) = 𝑒^(−|𝑥|) as a function in Matlab.

main.m
distribution = @(x) exp(-abs(x));
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2. Generate a large number of random samples from this distribution using the random function in Matlab.

main.m
numSamples = 1000000; % Choose the number of samples
samples = random('exp', 1, numSamples, 1); % Generate random samples from exponential distribution
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3. Calculate the value of 𝑥^2 for each sample.

main.m
squaredSamples = samples.^2;
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4. Compute the mean value of 〈𝑥^2〉 by taking the average of the squared samples.

main.m
meanValue = mean(squaredSamples);
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The variable meanValue will contain the estimated mean value of 〈𝑥^2〉 using random sampling with the given distribution 𝜋(𝑥) = 𝑒^(−|𝑥|).

Note that the expectation value of 〈𝑥^2〉 for the given distribution 𝜋(𝑥) = 𝑒^(−|𝑥|) can be obtained analytically and is equal to 2.

References:

• MATLAB Documentation: random
• MATLAB Documentation: mean