Computer program: As discussed during the class, let the random variable Y= X₁+X2+ ... +Xn, where Xi's are identically and independently distributed (i.i.d) random variables. The PDF of Y can be obtained by convolving all Xi's PDFs. If n becomes large (say, 30) the pdf of Y tends to the Gaussian Density. Let us see it for ourselves. Let us use the uniform density for Xi's PDF. Uniform pdf is far from Gaussian. Use Python to generate uniform density (10 samples between -1 and 1. It is discrete pdf as we cannot plot continuously using computer. But it is OK. (note: in Matlab it is Conv (u, v), where u and v are vectors (in this case starting with 10 elements each. You need to find equivalent function in Python.) Next round, you will need to increase the data size to 20 each as the convolution continues to expand each time, and so on. First take convolution of two uniform pdf, then, add one more uniform pdf (convolving the previous result with this new uniform pdf). Continue this by plotting each result to see if at what n, the result looks like a Gaussian density.
Computer program: As discussed during the class, let the random variable Y= X₁+X2+ ... +Xn, where Xi's are identically and independently distributed (i.i.d) random variables. The PDF of Y can be obtained by convolving all Xi's PDFs. If n becomes large (say, 30) the pdf of Y tends to the Gaussian Density. Let us see it for ourselves. Let us use the uniform density for Xi's PDF. Uniform pdf is far from Gaussian. Use Python to generate uniform density (10 samples between -1 and 1. It is discrete pdf as we cannot plot continuously using computer. But it is OK. (note: in Matlab it is Conv (u, v), where u and v are vectors (in this case starting with 10 elements each. You need to find equivalent function in Python.) Next round, you will need to increase the data size to 20 each as the convolution continues to expand each time, and so on. First take convolution of two uniform pdf, then, add one more uniform pdf (convolving the previous result with this new uniform pdf). Continue this by plotting each result to see if at what n, the result looks like a Gaussian density.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 29CR
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We are given that Y = X1 + X2 + X3 + ..... Xn. Start with a uniform distribution density for Xi that has 10 (10 is the n here) values between -1 and 1. This will the
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