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Test 2 for ISA5305, Fall 2009
Subject: Probability Theory
1. (10%) If two dice are rolled, what is the probability that the sum of upturned
faces will equal 7 and 8, respectively?
2. (10%) A foreign student club lists as its members 2 Canadians, 3 Japanese, 5
Italians, and 2 Germans. If a committee of 4 is selected at random, find the
probability that
(a) all nationalities are represented;
(b) all nationalities except the Italians are represented.
3. (10%) When coin A is flipped comes up heads with probability 1/4, whereas
when coin B is flipped it comes up heads with probability 3/4. Suppose that one
of these coins is randomly chosen and it flipped twice. If both flips land heads,
what is the probability that coin B was the one flipped?
4. (20%) Write down the following probability density function and the moment
generating function given each of the following conditions.
(a) Binomial distribution with mean 60 and variance 24.
(b) Exponential distribution with variance 4.
(c) Poisson distribution with variance 4.
(d) Normal distribution with mean 3 and variance 4.
(e) 2 distribution with degrees of freedom 6.
5. (10%) Let {X,Y,Z} be a random sample of size 3 from a Poisson distribution with
mean 2, define W=X+Y+Z.
(a)
(b)
(c)
(d)
What are the moment-generating functions MX(t) and MW(t), respectively?
Name the distribution of W.
What is E(W), the expectation of the random variable W?
What is Var(W), the variance of W?
6. (10%) Suppose there are three defective items in a lot of 50 items. A sample of
size 10 is taken at random without replacement. Let X denote the number of
defective items in the sample. Find the probability that the sample contains
(a) Exactly one defective item.
(b) At most one defective item.
7. (10%) X1, X2, …, Xn are independent continuous random variables with means μ1,
μ2,…, μn and variances  12 ,  22 ,,  n2 , respectively. Let a random variable Y
be defined by Y  a1 X 1  a 2 X 2    a n X n , where a1 , a2 ,, an are real
numbers. (a) Compute the mean and variance of the random variable Y. (b)
1
Assume the moment generating function for Xi is exp(  i t   i2 t 2 ) for
2
i=1, …, n. Derive the moment generating function for Y.
8. (10%) Let X~N(0,1) be normally distributed with zero mean and unit variance, i.e.
its probability density function is given by
1 
f X ( x) 
e
2
x2
2
A random variable Y is defined to be Y = X2. What is the probability density
function for Y? Show your derivation.
9. (10%) Let Z=[X, Y]t be a random vector having a multivariate normal
distribution with the probability density function listed below
f(x,y)=(1/12π)exp[-{9(x-2)2+4(y-1)2}/72]
What is the mean vector and covariance matrix of Z?