Reference no: EM132831935

Assignment

Question 1. A random variable X has probability density function

f (x) = 2x for 0 < x < 1

and f (x) = 0 otherwise.

(a) Show that μ = E(X) = 2/3 and σ² = Var(X) = 1/18.

(b) If X₁, X₂, . . . are independent and identically distributed with this probability density function, then use the Central Limit Theorem to find an approximation to

Pr(∑⁴⁵⁰_i=1) X_i ≥ 310.

(c) Show that Var(X²) = 1/12.

(d) Show that

1/n ΣX_i² P_→ 1/2 as n -→ ∞

Question 2. Suppose that two independent random samples of size n₁ and n₂ observations are obtained. Let X₁, X₂, . . . , X_n1 and Y_1, Y₂, . . . , Y_n2 be the two random samples and suppose that E(X_i) = E(Y_i) = λ and Var(X_i) = Var(Y_i) = σ² < ∞. The sample means from each sample are as follows,

X^- = Σⁿ¹_i=1 X_i/n₁ Y^- = Σⁿ²_i=1 Yi/n₂

Also defined two pooled estimators,

L₁ = n₁X^- + n₂Y^-/n₁+n₂

and L₂ = 1/2 (X^- + Y^- ).

(a) Show that both of the pooled estimators E(L₁) = E(L₂) = λ.

(b) Find Var(L₁) and Var(L₂).

(c) Show that both estimators are consistent estimators of λ.