Download STAT_HW3

STATISTICS
Due Dec. 11, 2007
Homework 3
1. Suppose random variables X and Y are jointly distributed with joint pdf
f X ,Y ( x, y )  ( x  y ) I ( 0,1) ( x) I ( 0,1) ( y ) .
(1) Find fY | X ( y | x) and FY | X ( y | x) .
(2) Calculate EY | X  x .
2. If random variable X has a Bernoulli distribution with parameter p, i.e.
( PX 1  p ), EY | X  0  1 , and EY | X  1  2 , what is EY  ?
3. Let f X ,Y ( x, y)  e( x  y ) I (o,) ( x) I (0,) ( y)
(1)
(2)
(3)
(4)
Find P1  X  Y  2 .
Find PX  Y | X  2Y .
Find P0  X  1 | Y  2.
Find m such that PX  Y  m  0.5 .
4. Let the joint density function of X and Y be given by f X ,Y ( x, y )  8 xy for
0<x<y<1 and be 0 elsewhere.
(1) Find EY | X  x .
(2) Find EXY | X  x .
(3) Find VarY | X  x
5. Suppose that X1,…, Xn are independent Bernoulli random variables; that is,
P[Xi = 1] = p, and P[Xi = 0] = 1p.
n
(1) Show that
X
i 1
i
has a binomial distribution with parameters n and p. [Hint:
n
Derive the moment generation function of
X
i 1
i
.]
2

(2) Calculate P  X n   .
n

6. Let Xi denote the number of meteors that collide with a test satellite during the ith
n
orbit. Let S n   X i ; that is, Sn is the total number of meteors that collide with
i 1
the satellite during n orbits. Assume that the Xi’s are independent and identically
distributed Poisson random variables having mean .
(1) Find E[Sn] and Var[Sn].
(2) If n =100 and  = 4, find approximately P[S100 > 400].
7. How many light bulbs should you buy if you want to be 95% certain that you will
have 1000 hours of light if each of the bulbs is known to have a lifetime that is
(negative) exponentially distributed with an average life of 100 hours? Assume
that one bulb is used until it is burnt out and then it is replaced, etc.
8. Suppose that X 1 and X 2 are means of two random samples of size n from a
population with variance  2 . Determine n such that the probability will be about
0.01 that the two sample means will differ by more than  . [Hint: Consider
Y  X1  X 2 .]
9. A research worker wishes to estimate the mean of a population using a sample
large enough that the probability will be 0.95 that the sample mean will not differ
from the population mean by more than 25% of the standard deviation. How large
a sample should he take?
10. Let X1 and X2 be a random sample from N(0,1).
(5) What is the distribution of
 X 2  X1 /
(6) What is the distribution of
 X 1  X 2 2 /  X 2  X1 2 ?
(7) What is the distribution of
 X1  X 2 /  X1  X 2 2 ?
2?