APPM 4/5520 Problem Set Four (Due Wednesday, October 5th) 1. Let X1 , X2 , . . . , Xn be a random sample from any distribution with mean and variance 2 . From class, we know that = X is an unbiased estimator of . ^ Suppose that we want to estimate the var

APPM 4/5520 Problem Set Three (Due Wednesday, September 14th) 1. Let X geom0 (p). (a) Compute the moment generating function of X. Be sure to give and explain restrictions, if any, on the domain of the mgf. (b) Let Y geom1 (p). How does Y relate to X? Use

APPM 4/5520 Problem Set Two (Due Wednesday, September 7th) 1. Suppose that X1 , X2 exp(rate = ). Find the distribution of 2. Let X1 and X2 have the joint pdf fX1 ,X2 (x1 , x2 ) = 2e-x1 -x2 I(0,x2 ) (x1 )I(0,) (x2 ). (a) Find the marginal pdfs for X1 and X

APPM 4/5520 Solutions to Problem Set One 1. The binomial random variable X counts the number of successes in n trials where p is the probability of success on any single trial. So, Y = n - X represents the number of failures in n trials. If we rename "fai

APPM 4/5520 Problem Set One (Due Wednesday, August 31st) 1. Let X be a random variable with the binomial distribution with parameters n and p, (ie: X bin(n, p). Find the distribution of Y = n - X. (Name it!) 2. Let X unif (0, 1). Find the distribution of

1 Introduction to Operations Research
In its first section, this introductory chapter first introduces operations research as a discipline. It defines its function and then traces its roots to its beginnings. The second section highlights some of the main

APPM 4/5520 Problem Set Nine (Due Wednesday, November 30th) 1. Let X1 , X2 , . . . , Xn be a random sample from the exponential distribution with rate . (a) Find the MLE (maximum likelihood estimator) for the "tail probability" P (X1 > x) for some fixed x

APPM 4/5520 Problem Set Eight (Due Wednesday, November 16th) 1. Let X1 , X2 , . . . Xn be a random sample from the binomial distribution with parameters m and p. Find the MLE of P (X = 0). 2. Let X1 , X2 , . . . , Xn be a random sample from the P oisson()

APPM 4/5520 Solutions to Problem Set Seven 1. The first population moment is
1 = E[X] =
x e-(x-) dx = + 1.
(Note that since the pdf is that of an exponential with rate 1 that has been shifted by to the right, the mean is 1 (the mean of the exponential wit

APPM 4/5520 Problem Set Seven (Due Wednesday, November 9th) 1. Consider the "shifted" rate 1 exponential distribution with pdf f (x) = e-(x-) I(,) (x). This distribution is frequently encountered in reliability and product lifetime analyses. The new param

APPM 4/5520 Solutions to Problem Set Six 1. Recall that we define a t-distribution with n degress of freedom by letting Z N (0, 1) and W 2 (n) be independent random variables. Let T = Then T t(n). For this problem, note that X1 X1 = = |X2 | 2 X2 X1 / = (X

APPM 4/5520 Problem Set Six (Due Wednesday, October 19) 1. Let X1 , X2 be a random sample from the N (0, 2 ) distribution. Find the distribution of X1 ? |X2 | Name it! (Hint: |X2 | =
2 X2 .)
2. Let X(1) , X(2) , . . . , X(n) denote the order statistics of

APPM 4/5520 Solutions to Problem Set Five 1. (a)
n i=1 (xi
- )2 = = =
n i=1 (xi n i=1
- x + x - )2
(xi - x)2 + 2(xi - x)(x - ) + (x - )2 - x)2 + 2(x - )
n n n i=1 (xi
n i=1 (xi
- x) +
n i=1 (x
- )2
For the middle term, note that
n i=1 n
(xi - x) =
n i=1
i

APPM 4/5520 Problem Set Five (Due Wednesday, October 12th) 1. Suppose that X1 , X2 , . . . , Xn N (, 2 ). (a) Show that
n 2 i=1 (xi - ) 2 n i=1 (xi - 2 iid
=
x)2
+
n(x - )2 . 2
(b) Write down the joint pdf for X1 , X2 , . . . , Xn and use part (a) to rewr

APPM 4/5520 Solutions to Problem Set Three 1. (a) f (x) = p(1 - p)x Icfw_0,1,2,. (x) MX (t) = E[etX ] = = p
x=0 tx x=0 e
p(1 - p)x
x
et (1 - p)
This geometric sum converges if et (1 - p) < 1. (Note that it is already positive since et > 0 and since 0 p