Description
Directions: Do all the problems. The total grade is 40 points, with all problems being 5 points each. You may work with others in the class or use any sources you like to study these problems, but when you go to write the solutions you should be able to write it using only the textbook and course notes. If you obtain facts, advice, or inspiration from any source outside the textbook or course notes, you must indicate in your solution what the source is. In all cases, you must make it clear to the reader what logic/work was needed to reach your solution. Solutions must be posted to Canvas– any legible version, handwritten or typed, is acceptable.
#1
Consider ten cards in a bag, labeled in order from 1 to 10.
(a) Draw three cards randomly and independently, with replacement. That is, 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 re- placement.
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#2
This exercise concerns truncated discrete distributions. If the random vari- able X has range {0, 1, 2, …}, we might define the 0-truncated random vari- able XT has pmf
P(XT = x) = P(X = x),x = 1,2,3,…. P(X > 0)
Write the pmf, mean, and variance of XT when (i) X ∼ P oi(λ) with λ > 0 and (ii) X has the pmf
x+r−1 r x
fX(x)= x p (1−p) ,x∈{0,1,2,…},
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?
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#5
Suppose the random variable T is the length of life of an object. Define the hazard function hT (t) of T to be
hT(t)=limP(t≤T <t+δ|T ≥t). δ→0 δ
If T is a continuous random variable, then
hT(t)= fT(t) =−d log(1−FT(t)).
1−FT(t) dt Verify the following indicated hazard functions.
(a)IfT ∼Exp(β)forsomeβ>0,thenhT(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)IfT ∼Logistic(μ,β)forsomeμ∈Randβ>0,thatis
FT (t) = 1 , 1 + e−(t−μ)/β
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 fX′ (x) + xfX(x) = 0. (ii) Show for x > 0 that
x−1 −x−3 < 1−FX(x) < x−1 −x−3 +3x−5. fX (x)
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#7
This exercise concerns the folded normal distribution. Let X have pdf 2 −x2/2
fX (x) = √2π e , x > 0. (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).
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