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.
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.
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 PDF for our Xi. The PDF of Y is where we convolute all of the PDFs for Xi. When graphed, the PDF of Y should look similar to a Gaussian density. Write the code in Python. I will upvote if it is correct.
Transcribed Image Text: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 X{'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 X{'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.
Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.
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