[SOLVED] stats510 - Homework 6

Description

#1

Suppose that 26 percent of the students in a certain high school are freshmen, 24 percent are sophomores, 30 percent are juniors, and 20 percent are seniors.

• If 15 students are sampled with replacement at random from the school,what is the probability that at least eight will be either freshmen or sophomores?
• Following part a, let X₃ denote the number of juniors in the 15 sampled students and X₄ be the number of seniors in that sample. Compute E(X₃ − X₄) and Var(X₃ − X₄).

#2

Suppose we roll two (potentially unfair) six-sided dice A and B so that the outcomes from A and B are independent. Let S be the sum of the two rolls, supported on {2,3,…,12}. Can the dice be constructed so that P(S = s) = 1/11 for each s ∈ {2,3,…,12}? Prove your answer.

#3

Let X₁,…,X_n be iid random variables, where X_n and S_n² are the sample mean and sample variance of the sequence.

• Show that

.

For the next two parts, assume that θ₁ = E[X_i] and θ_j = E[(X_i − θ₁)^j] are finite. Note that θ₁ is the mean and θ_j are the second, third, and fourth centered moments.

• Show that Var(.
• Find) as a function of θ₁,θ₂,θ₃,θ₄. When does equal 0?

#4

Suppose that X₁,…,X_n are iid Exp(1) random variables. Recall that X₍₁₎ = min(X₁,…,X_n) and X_(n) = max(X₁,…,X_n). Determine the conditional pdf of X(1) given X(n) = yn.

#5

Let X₁,…,X_n be iid Exp(λ) random variables and Y_n = X_(n) − logn = max(X₁,…,X_n) − logn.

• Show that Y_n converges in distribution to some random variable Y_∞; write the pdf of Y_∞.
• Find the mgf and expectation of Y_∞. For this, it will be useful to consider the positive constant

 .

You may use without proof that

γ = −Γ⁰(1),

where we recall the gamma function at each positive real number t is

(This γ is the Euler-Mascheroni constant and is approximately equal to 0.577; it appears in many places in analysis and probability.)

#6

• Let p > 0 be arbitrary and assume E|X_n|^p → 0 as n → ∞. Show that

X_n converges in probability to 0.

• Let X₁,X₂,… be uncorrelated random variables such that E[X_i] = µ for all i and Var[X_i] ≤ C < ∞ for some C independent of i. Show that the

sample average X_n converges in probability to µ.

#7

In this exercise, we consider improvements to the Chebyshev inequality and the law of large numbers for select examples.

• Let X be a random variable with moment generating function M_X(s), defined in a neighborhood of 0. Show for every real number t that

P.

• Let X ∼ Bin(n,0.5). Find a number b > 1 such that

converges to a positive constant as n → ∞.

#8

A physicist makes 25 independent observations of the specific gravity of a given body. Each measurement has a standard deviation of σ.

• Using the Chebyshev inequality, find a lower bound for the probabilitythat the average of their measurements differs from the actual specific gravity by less than σ/
• Use the central limit theorem to approximate the probability in part (a).