#1
Consider ten cards in a bag, labeled in order from 1 to 10.
(a) 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.
(b) 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 XT has pmf
P(X = x)
P(XT = x) = ,x = 1,2,3,.... P(X > 0)
Write the pmf, mean, and variance of XT 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
fn(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 hT (t) of T to be
.
If T is a continuous random variable, then
.
Verify the following indicated hazard functions.
(a) If T ∼ Exp(β) for some β > 0, then hT (t) = 1/β.
(b) If S ∼ Exp(β) and T = S1/γ for some β,γ > 0, then hT (t) = (γ/β)tγ−1. (c) If T ∼ Logistic(µ,β) for some µ ∈ R and β > 0, that is
,
then hT (t) = FT (t)/β.
#6
Let X be an N(0,1) distribution and let fX(x) be its pdf and FX(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
.
(a) Find the mean and variance of X.
(b) 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
fX,Y (x,y) = 2(x + y),0 ≤ x ≤ y ≤ 1.
Write the marginal pdfs fX(x) and fY (y).
