Q2)

Generate the empirical PDF (histogram) of IZI and IZI2 by generating n realization of X and Y where X and Y are i.i.d. Gaussian random variables of zero mean and unit variance, and obtain n realization of Z and |Zl2 via Z -X +j Y. Plot the histogram of these samples for n-100 and compare with the theoretical PDF of Rayleigh distribution and exponential distribution. Do they agree? Hint: you need to scale the histogram in such a way that it becomes an empirical PDF. Hints: If X1 and X2 are independent random variables uniformly distributed over [0,1], then are independent Gaussian random variables, each with zero mean and unit variance. Vary the variance of X and Y as 2-1, 3, and 5, repeat Q2 for n-1000. Comment on the result Hint: A standard normal random variable, i.e., zero mean and unit variance, can be converted to a random variable of desired mean and variance 2, namely x~ N , 2) , by the transformation: x + . Xstd-normal Generate the empirical PDF (histogram) of IZI and IZI2 by generating n realization of X and Y where X and Y are i.i.d. Gaussian random variables of zero mean and unit variance, and obtain n realization of Z and |Zl2 via Z -X +j Y. Plot the histogram of these samples for n-100 and compare with the theoretical PDF of Rayleigh distribution and exponential distribution. Do they agree? Hint: you need to scale the histogram in such a way that it becomes an empirical PDF. Hints: If X1 and X2 are independent random variables uniformly distributed over [0,1], then are independent Gaussian random variables, each with zero mean and unit variance. Vary the variance of X and Y as 2-1, 3, and 5, repeat Q2 for n-1000. Comment on the result Hint: A standard normal random variable, i.e., zero mean and unit variance, can be converted to a random variable of desired mean and variance 2, namely x~ N , 2) , by the transformation: x + . Xstd-normal