Question

Define X1 = Z1, X2 = 22, ..., Xn = Zn and X = 36 L3fXi.

Answer 1

(d)

X1, X2,.., Xn are the squares of standard normal variables. Thus,

X_i follows Chi Square distribution with 1 degree of freedom.

E[X_i] = 1 and Var[X_i] = 2

By Central limit theorem,

$\bar{X}$ ~ N($\mu$ = E[X_i] , $\sigma^2$ = Var[X_i] /36)

or,

$\bar{X}$ ~ N($\mu$ = 1, $\sigma^2$ = 2/36)

$\sigma$ =
V2/36
= $\sqrt{2}$ / 6

A = P(| 8 – 11<V2/3)=P(-V2/3<Ě -1<V2/3)

=P(-2< (7 - 1)/(V2/6) <2)

= P(-2<Z < 2)
  
(1 - 1)/(V2/6) ~ N(0,1)

= P(Z < 2) - P(Z < -2)

= 0.9772 - 0.0228

= 0.9544

(e)

By Chebyshev's inequality,

P(X – E[X]]>b) < Var(X)/64

Using b = $\sqrt{2}$ /3

P(|X – 11 2 V2/3) < (2/36)/(V2/3)?

P(|X – 11 2 V2/3) < 0.25

1 - P(|X – 11 < V2/3) < 0.25

P(x - 1<V2/3) 21 – 0.25

p(\X – 11 <V2/3) > 0.75

Lower Bound for A is 0.75