Question

Question 2. Consider a random variable X~x20 - Answer the following. i) ii) What are E(X) and Var(X)? What value is the value xão,0.1 that satisfies P(xão > xão,0.1) = 0.1? Suppose you draw random samples of n=40 individual values from a population where the relative frequency of values follows a Chi-square distribution with 20 degrees of freedom. That is Xi~xão for every random variable in the random sample (X1 , X2 ,... X40). iii) iv) Use the Central Limit Theorem to find the value P(X > x*) = 0.1. What would you expect would happen to the value x* as the sample size n increases? Why?

Answer 1

ANSWER:

given that:

Consider a random variable X $\sim$ X²₂₀

let US consider,

A random variable X $\sim$ X²₂₀

let,

X fallow chi - square distribution with 20 degree of freedom ( df ) .

(i) What are E (X) $ Var ( X )

we know that,

if X follows chi- square with n degree of freedom , then

It's mean is given as n $ variable as 2n.

E [ X ] = 20

Var ( X ) = 40

( ii ) what value is the value X_20,0.1² that catisties P ( X²₂₀  > X_20,0.1²  ) = 0.1

For 20 df $ e

10 % ( 0.1) level of significance.

we have from the chi-square table.

$\Rightarrow P(X_2>28.4)=0.1$

so,

our required value is 28.4

(iii) use the central limit theorem to find the value  $\Rightarrow P(\bar{X}>x)=0.1$

we have ,

$E[X_{i}]=20$

Var X; = 40

so ,

$Var[\bar{X}]=\frac{Var(X)}{N}=\frac{40}{40}=1$

$Var[\bar{X}]=1$

Now by using

Central limit theorem:

$Z=\frac{\bar{X}-E[X]}{\sqrt{Var(\bar{X})}}\sim N(0,1)\, \, as\, \, \, n\rightarrow \infty$

$Z=\frac{\bar{X}-20}{1}$

We know that,

$\therefore X^{*}=1.28$

( i V ) wthat would you expect would happen to the value x* as the sample size n increales ? why

Due to,

Central limit theorem :

There won't be signifficant change in the value of Z as n is increases

Hence ,

The value of X^* = 1.28

will remain same as n increaser.