Assume that we have uniform random number generator (URNG) that is well

modeled by a sequence of i.i.d. uniform RVs Xi, i = 1, . . . , n, where Xi is the ith

output of the URNG. Assume that

fXi (Xi) =

1

a

rect

2

xi ? a/2

a

3

.

(a) Show that with Zn = X1 + . . . + Xn, E[Zn] = na/2. (b) Show that Var(Zn) =

na2/12. MATLAB program that computes the plots fZn(z) for n =

2, 3, 10, 20. MATLAB program that plots Gaussian pdf's N

.

na

2 , na2

12

/

for n = 2, 3, 10, 20 and compare fZn(z) with N

.

na

2 , na2

12

/

for each n. (e) For each

n compute P[?n ? k?n ? Zn ? ?n + k?n], where ?n = na/2, ?2

n = na2/12 for

a few values of k, for example, k = 0.1, 0.5, 1, 2, 3. using both fZn(z) and

N

.

na

2 , na2

12

/

. Choose any reasonable value of a, for example, a = 1.

\f