MATH 357: Assignment 1
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MATH 357: Assignment 1
Winter 2023
Problem 1. Suppose X1 , X2 , . . . are continuous and identically distributed random variables with pdf and cdf f and F , respectively. Each Xi represents annual rainfall at a given location or region in year i.
(a) Find the distribution of the number of years until the first year’s rainfall X1 is exceeded for the first time.
(b) Show that the expected value of the number of years until the first year’s rainfall X1 is exceeded for the first time is o.
Problem 2. Let X be a real-valued random variable with an unknown distribution F , where F (x) = P (X < x), Ax e R. Let X1 , X2 , . . . , Xn be a random sample of size n from F . Recall the empirical cdf (ecdf) Fn (x) discussed in class which is used as an “educated guess” of F (x), for any x e R. Assume that we are also interested in the general probabilities P (X e B), for any Borel set B, which is called the probability distribution of X . Use the above random sample to answer the following questions.
╱ 、
(b) By extending the definition of the ecdf Fn (x), provide an “educated guess” for the probability distribution of X and call it Pn (B), for any Borel set B .
(c) Find the expected value and variance of Pn (B).
(d) Choose an and bn such that the limiting distribution of an {Pn (B) _ bn } is N(0, 1), as n → o.
Problem 3. Let X1 , X2 , . . . , Xn be a random sample of size n from a distribution F with a pdf f . Consider the two order statistics X(1) = min1一i一n Xi and X(n) = max1一i一n Xi , for any n > 1. The sample range is defined as Rn = X(n) _ X(1) .
(a) Derive the joint cdf and pdf of (X(1) , X(n)) in terms of f and F . (b) Find the cdf and pdf of the sample range Rn .
(c) For α > 0, assume that limz≤→ xa P (X1 > x) = b > 0. Find the limiting distribu- tion (both cdf and pdf) of Un = (bn)│1/aX(n), as n → o.
(d) Assume that limz≤→ ez P (X1 > x) = b > 0. Find the limiting distribution (both cdf and pdf) of Vn = X(n) _ log(bn), as n → o.
Problem 4. Let X1 , X2 , . . . , Xn be a random sample of size n from N(µ, σ2 ).
(a) Show that Yi =/xi│口u 、2 ~ χ, for i = 1, . . . , n.
(b) Using the result in part (a), show that
n
Yi ~ χn) .
i=1
(c) In class, using a transformation technique we proved that the sample mean n and the sample variance Sn(2) are independent random variables (statistics). Verify the same claim using the moment generating technique.
Problem 5. Assume that X1 , X2 , . . . , Xm and Y1 , Y2 , . . . , Yn are two independent ran- dom samples from N (µ1 , σ1(2)) and N (µ2 , σ2(2)), respectively.
(a) Find the distribution of m _ n .
(b) Find the distribution of U (n, m) = (m口(│))sm(2) + (n)sn(2) .
(c) Find the constant c(n, m) > 0 such that
c(n, m) {m _ n _ (µ1 _ µ2 )}
^U (n, m)
The constant c(n, m) > 0 is also a function of (σ1(2), σ2(2)).
(d) Assume that X1 , X2 , . . . , Xm and Y1 , Y2 , . . . , Yn are two independent random sam- ples from, respectively, the distributions F1 and F2 , such that µ 1 = E(X1 ), σ1(2) = Var(X1 ) < o and µ2 = E(Y1 ), σ1(2) = Var(Y1 ) < o. Without using the t distri- bution in part (a), directly find the limiting distribution of the random sequence V (n, m), as both (m, n) → o. Note that you still need your choice of c(n, m) from
part (c) to answer this part.
Problem 6.
(a) Let X ~ t(ν). Show that X2 ~ F (1, ν).
(b) Let Y ~ F (p, q). Show that,
(i) Y │1 ~ F (q, p).
(ii) ~ Beta(p/2, q/2), where Beta is the beta distribution and r = p/q .
Note: the pdf of a Y ~ Beta(a, b) distribution:
f (y; a, b) = y(a│1)(1 _ y)(b│1) , 0 < y < 1
where Γ(.) is the gamma function, and a, b > 0 two parameters of a Beta distribution.
2023-01-28