# Gaussian random variable with PDF

• February 7th 2009, 08:17 AM
Bob001
Gaussian random variable with PDF
Hi, I had a question I am hoping someone can give me some pointers on.

Question: Guaussian random variable X with PDF, fx(x)=1/(sqrt(2*pi))*e^(-x^2/2) ; -infinity< x < +infinity.

a) The Gaussian is passed through a square law device with output Y=X^2. Find the PDF of Y.
b) THe Gaussian is "passed through a full wave rectifier with output Y=absolutevalue(X). Find the PDF of Y
*See attached image for hint given

Second part to question: Consider the three level limiter shown in the figure (attached)

a)Find the PDF of Y if X is the Gaussian PDF of the previous problem. Express answer in terms of the CDF of X given by phi(x)=integral from t=-infinity to t=x of 1/(sqrt(2*pi))*e^(-t^2/2) dt
b)Sketch both the PDF , fY(y) and the CDF, FY(y) for the random variable Y.
Thanks for whatever help I can get!
• February 7th 2009, 03:05 PM
mr fantastic
Quote:

Originally Posted by Bob001
Hi, I had a question I am hoping someone can give me some pointers on.

Question: Guaussian random variable X with PDF, fx(x)=1/(sqrt(2*pi))*e^(-x^2/2) ; -infinity< x < +infinity.

a) The Gaussian is passed through a square law device with output Y=X^2. Find the PDF of Y.
[snip]

• February 7th 2009, 03:10 PM
mr fantastic
Quote:

Originally Posted by Bob001
Hi, I had a question I am hoping someone can give me some pointers on.

Question: Guaussian random variable X with PDF, fx(x)=1/(sqrt(2*pi))*e^(-x^2/2) ; -infinity< x < +infinity.

[snip]

b) THe Gaussian is "passed through a full wave rectifier with output Y=absolutevalue(X). Find the PDF of Y
*See attached image for hint given

[snip]

Use the same technique as (a). Here's a start:

The cdf of Y is $F(y) = \Pr(Y < y) = \Pr(|X| < y) = \Pr(-y < X < y) = 2 \Pr(0 < X < y)$.

Now get into the calculation.
• February 7th 2009, 05:20 PM
oswaldo
Nice work Mr. Fantastic.
For the first part X^2 should be standard Chi-square distibution. Should not be hard to drive its pdf algebrically.

-O
• February 8th 2009, 04:46 PM
Bob001
Thank you for the tips!