[SOLVED] stats510 - Homework 4

Description

#1

Consider ten cards in a bag, labeled in order from 1 to 10.

• Draw three cards randomly and independently, with replacement. Thatis, any card drawn is put back in the bag. Write the expectation of the sum of the three numbers shown.
• Same problem, except the three cards are drawn randomly without replacement.

#2

This exercise concerns truncated discrete distributions. If the random variable X has range {0,1,2,…}, we might define the 0-truncated random variable X_T has pmf

P(X = x)

P(X_T = x) = ,x = 1,2,3,…. P(X > 0)

Write the pmf, mean, and variance of X_T when (i) X ∼ Poi(λ) with λ > 0 and (ii) X has the pmf

,

with r a positive integer and p a real number in (0,1).

#3

A population of N animals has had a number M of its members captured, marked, then released into the original population. Let X be the number of animals that are necessary to recapture (without re-release) in order to obtain K marked animals. Write the pmf, expectation, and variance of X.

#4

Let H and T be independent Poisson random variables with parameters λ,µ > 0, respectively. (a) Show that H + T ∼ Poi(λ + µ). (b) Show that the conditional distribution given any integer n ≥ 1, given by

f_n(x) = P(H = x|H + T = n),x ∈ {0,1,…,n},

follows a binomial distribution with parameters n and p, for some p. What is p?

#5

Suppose the random variable T is the length of life of an object. Define the hazard function h_T (t) of T to be

.

If T is a continuous random variable, then

.

Verify the following indicated hazard functions.

• If T ∼ Exp(β) for some β > 0, then h_T (t) = 1/β.
• If S ∼ Exp(β) and T = S^1/γ for some β,γ > 0, then h_T (t) = (γ/β)t^γ−1. (c) If T ∼ Logistic(µ,β) for some µ ∈ R and β > 0, that is

,

then h_T (t) = F_T (t)/β.

#6

Let X be an N(0,1) distribution and let f_X(x) be its pdf and F_X(x) be its cdf. (i) Show that ) = 0. (ii) Show for x > 0 that

.

#7

This exercise concerns the folded normal distribution. Let X have pdf

.

• Find the mean and variance of X.
• If X has a folded normal distribution, find the transformation g and values α,β > 0 such that Y = g(X) has the Gamma(α,β) distribution.

#8

Let X,Y be continuous random variables whose joint pdfs is

f_X,Y (x,y) = 2(x + y),0 ≤ x ≤ y ≤ 1.

Write the marginal pdfs f_X(x) and f_Y (y).