Directions: Do all the problems. The total grade is 40 points, with all problems being worth 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
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.
(a) 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 sopho- mores?
(b) 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).
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#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,…,Xn be iid random variables, where Xn and Sn2 are the sample mean and sample variance of the sequence.
(a) Show that
1 n
Sn2 = 2n(n−1) (Xi −Xj)2.
i,j =1
For the next two parts, assume that θ1 = E[Xi] and θj = E[(Xi −θ1)j] are finite. Note that θ1 is the mean and θj are the second, third, and fourth centered moments.
(b) Show that Var(S2) = 1 θ − n−3θ2. n n 4 n−12
(c) Find Cov(Xn, Sn2) as a function of θ1, θ2, θ3, θ4. When does Cov(Xn, Sn2) equal 0?
#4
Suppose that X1, …, Xn are iid Exp(1) random variables. Recall that X(1) = min(X1, …, Xn) and X(n) = max(X1, …, Xn). Determine the conditional pdf of X(1) given X(n) = yn.
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#5
Let X1, …, Xn be iid Exp(λ) random variables and Yn = X(n) − log n = max(X1, …, Xn) − log n.
(a) Show that Yn converges in distribution to some random variable Y∞; write the pdf of Y∞.
(b) Find the mgf and expectation of Y∞. For this, it will be useful to consider the positive constant
n1 γ:=lim −log(n) .
k
You may use without proof that
γ = −Γ′(1),
where we recall the gamma function at each positive real number t is
it appears in many places in analysis and probability.)
#6
(a)Letp>0bearbitraryandassumeE|Xn|p →0asn→∞. Showthat
Xn converges in probability to 0.
(b) Let X1, X2, … be uncorrelated random variables such that E[Xi] = μ for all i and Var[Xi] ≤ C < ∞ for some C independent of i. Show that the sample average Xn converges in probability to μ.
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n→∞
k=1
∞ x=0
xt−1e−x dx.
(This γ is the Euler-Mascheroni constant and is approximately equal to 0.577;
Γ(t) =
#7
In this exercise, we consider improvements to the Chebyshev inequality and the law of large numbers for select examples.
(a) Let X be a random variable with moment generating function MX(s), defined in a neighborhood of 0. Show for every real number t that
P(X ≥ t) ≤ mine−stMX(s). s>0
(b) Let X ∼ Bin(n, 0.5). Find a number b > 1 such that n X 1 1
bP n−2≥10
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.
(a) Using the Chebyshev inequality, find a lower bound for the probability that the average of their measurements differs from the actual specific gravity by less than σ/4.
(b) Use the central limit theorem to approximate the probability in part (a).
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