1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi....

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Question

1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi. (a) Use CLT to get a large sample distribution of Y (b) For n = 100, give an approximation for P(Y > 100) (c) Let X be the sample mean, then approximate P(1.1 < X < 1.2) for n = 100.

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Answer 1

Answer #1

a)

$E(Y) = E(sum_{i=1}^{n}X_i) = sum_{i=1}^{n}E(X_i) = sum_{i=1}^{n}1 = n*1 = n$

and Var(Y) Var Xi)- Var(Xi) i=1 i=1 i=1

and sd(Y) = sqrt{Var(Y)} = sqrt{n}

Using CLT, we can write -in ~ Normal(0, l)

For n = 100

Y10 V100 Normal(0,1) or Y10 Normal (0, 1) 10 or Y Normal (100, 102)

b) Y100 100-100 4 10 10

P(Y> 100)1- P(Z< 0)

Ply > 100)-1-4) ( 0 )

P(Y > 100) 1-0.50 = 0.50

c)

$E(ar{X}) = E(rac{1}{n}sum_{i=1}^{n}X_i) = rac{1}{n}sum_{i=1}^{n}E(X_i) = rac{1}{n}sum_{i=1}^{n}1 = 1$

and i=1 i=1

and T11

Using CLT, we can write X- 1 Normal(0,1) 7t

For n = 100

X-1 ~ Ņormal (0, 1) 100 or Normal(0, 1) 0.1

Now,

P(1.1 < ar{X} < 1.2) = P(ar{X}<1.2) - P(ar{X}<1.1)

0.1 0.1 0.1 0.1

Rightarrow P(1.1 < ar{X} < 1.2) = P(Z<2) - P(Z<1 )

Rightarrow P(1.1 < ar{X} < 1.2) = Phi(2) - Phi(1)

Rightarrow P(1.1 < ar{X} < 1.2) = 0.977250-0.841345

P(1.1 < X < 1.2) = 0.135905

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Answer 2

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1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi....

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