Math 532 Spring 2008
Test 1
Name:
x/2, if 0 x 2 1. Find Var(X) for the random variable X with density function f (x) = 0, Solution E(X) =
0 2 2
otherwise.
xf (x) dx =
1 2 1 2
2
x2 dx =
0 2
4 3
E(X 2 ) =
0
x2 f (x) dx =
x3 dx = 2
0
V ar(X) = E(X 2 ) - [E(X
Math 532
1.
Quiz 6
Name:
Let X and Y be independent random variables each uniformly distributed over (0, 1).
Find P (Y X |Y 1/2).
P (Y X |Y 1/2) =
2.
Let X and Y be continuous random variables with joint density function
for 0 x 1, x y 2x
0
f (x, y ) =
8
Math 532
Quiz 5
Name:
Suppose n balls are distributed at random into r boxes. Let Xi be dened by
1
Xi =
1.
(a)
if box i is empty,
0
otherwise.
Find E (Xi ).
E (Xi ) =
(b) For i = j , nd E (Xi Xj ).
E (Xi Xj ) =
2.
Let S denote the number of empty boxes. F
Math 532
Quiz 4
Name:
An experiment consists of five fair coins being tossed simultaneously. Repeated trials of the experiment are conducted until all five coins are heads. Let N be the number of trials required until all five coins are heads. 1. Calculat
Math 532
1.
Quiz 3
Name:
Let S = cfw_1 , 2 , 3 , P (1 ) = P (2 ) = P (3 ) = 1/3. Dene random variables X and Y by
X (1 ) = 1, X (2 ) = 2, X (3 ) = 3;
Y (1 ) = 2, Y (2 ) = 3, Y (3 ) = 1.
Let FX +Y (x) be the cumulative distribution function of X + Y . Fill
Math 532
Quiz 3
Name:
1.
Let S = cfw_1 , 2 , 3 , P (1 ) = P (2 ) = P (3 ) = 1/3. Dene random variables X and Y by X (1 ) = 1, X (2 ) = 2, X (3 ) = 3; Y (1 ) = 2, Y (2 ) = 3, Y (3 ) = 1. Let pX (x), pY (x), and pX +Y (x) be the probability distribution fun
Math 532
Quiz 2
Name:
1. A die is rolled three times. What is the probability that you get a larger number each time?
Answer 2. Two screws are missing from a machine that has screws of three dierent sizes. If three screws of dierent sizes are sent over, w
SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY
EXAM P PROBABILITY
EXAM P SAMPLE SOLUTIONS
Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society
Some of the questions in this study note are taken from past SOA/CAS examinations.
P-0
DISCRETE DISTRIBUTIONS
Uniform Distribution Bernoulli Trial: = cfw_success, failure. P (success) = p, P (failure) = q = 1 - p. Bernoulli Distribution B = B(1, p): P (B = 1) = p, P (B = 0) = 1 - p = q.
Binomial Distribution B(n, p): Repeat Bernoulli tria