Matlab Help. just a starter Create N random variables (X_1,..., X_n) each with L

Question

Matlab Help. just a starter

Create N random variables (X_1,..., X_n) each with L samples. These should be independent and identically distributed. Generate S_n, which will be the sum of these random variables. Find the mean and variance of X_1. i.e. mu and sigma^2. Since the RV's are iid's, mu will be the mean of all the other RV's in this set too. You can verify this by finding the mean of a few other variables. Create the random variable Z_n by adding N random variables as: Z_n = S_n - n mu / sigma Squareroot n Plot the pdf of Z_n

Explanation / Answer

Let us have L samples within each of X1, X2, X3, XN as given below:

X1: x11, x12, x13, .............x1L

X2: x21, x22, x23, .............x2L

X3: x31, x32, x33, .............x3L

.........................................................

XN: xN1, xN2, xN3, .............xNL

These are independent and identically distributed random variables with Mean x1', x2', x3'...........xn'

For independent and identically distributed randomvariable E(xi') = µ for Xi's and Var(xi') =

Now if Sn = (X1 + X2 +X3 +............ Xn ) / n

Mean = E ( Sn ) = E ( (X1 + X2 +X3 +............ Xn ) / n ) = (E(X!) + E(X2) + .......E(X3)) / n

= (µ +µ+µ+...........+µ) / n = nµ / n = µ

Var ( (X1 + X2 +X3 +............ Xn ) / n ) = 1/n2[Var (X1) + Var ( X2) + Var (X3) +.......Var (Xn)]

Since X1, X2, X3, ........, Xn are independent and identically distributed randomvariable

SO, Var ( (X1 + X2 +X3 +............ Xn ) / n ) = 1/n2 [ +++.......+] = 1/n2 *n* = /n

If X is a normal variate with mean µ and SD , the Z = (x-µ)/ follow normal distribution with mean 0 and standard deviation 1.

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