Problem 1 - Let X be a random variable taking on value -2 with probability 1/3 and value 1 with probability 2/3. Compute E(X), E(X^2), E(1/X), and var(X).
Problem 2 - Each Ferrero Kinder Egg (http:/en.wikipedia.org/wiki/Kinder_Surprise) conceals a toy.
Sam Chrisinger
2/13/14
MATH3100 Assignment 4
2.2
6. To estimate the percent of district voters who oppose a certain ballot measure,
a survey organization takes a random sample of 200 voters from a district. If
45% of the voters in the district oppose the
MTH135/STA104: Probability
Homework # 4
Due: Tuesday, Sep 27, 2005
Prof. Robert Wolpert
1. (from Prob 2 p. 121) Find Poisson approximations to the proabilities
of the following events in 1000 independent trials with probability 0.005 of
success on each tr
Homework Problem Set 4
1. Use a counting argument to prove that
2n
n
=2
+ n2
2
2
2. Use a counting argument to prove that
2
n
X
n
2n 1
k
=n
k
n1
k=0
3. Use a counting argument to prove that
n3 = 1 + 3(n 1) + 3(n 1)2 + (n 1)3
4. Use a counting argumen
Homework Problem Set 1
(1.1.5) 1. Consider the bet that all three dice will turn up sixes at least once in n rolls of three
dice. Calculate f (n), the probability of at least one triple-six when three dice are rolled
n times. Determine the smallest value
Homework Problem Set 7
(5.2.7) 1. Explain how you can generate a random variable whose cumulative distribution function is
0, if x < 0,
x2 , if 0 x 1,
F (x) =
1, if x > 1.
(5.2.9) 2. Let U , V be random numbers chosen independently from the interval [0,
Homework Problem Set 8
(6.1.6) 1. A die is rolled twice. Let X denote the sum of the two numbers that turn up, and
Y the difference of the numbers (specifically, the number on the first roll minus the
number on the second). Show that E(XY ) = E(X)E(Y ). A
Homework Problem Set 9
(6.3.3) 1. The lifetime, measure in hours, of the ACME super light bulb is a random variable T
with density function fT (t) = 2 tet , where = .05. What is the expected lifetime of
this light bulb? What is its variance?
(6.3.7) 2. Le
Homework Problem Set 6
(5.1.6) 1. Let X1 , X2 , . . . , Xn be n mutually independent random variables, each of which is
uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the
Xi s. Find the distribution of Y .
(5.1.7) 2. A die
Homework Problem Set 2
(2.2.1) 1. Suppose you choose at random a real number X from the interval [2, 10].
(a) Find the density function f (x) and the probability of an event E for this experiment, where E is a subinterval [a, b] of [2, 10].
(b) From (a),
Homework Problem Set 3
(3.1.6) 1. In arranging people around a circular table, we take into account their seats relative
to each other, not the actual position of any one person. Show that n people can be
arranged around a circular table in (n 1)! ways.
2
Homework Problem Set 12
(10.1.3) 1. Let p be a probability distribution on cfw_0, 1, 2 with moments 1 = 1, 2 = 3/2.
(a) Find its ordinary generating function h(z).
(b) Using (a), find its moment generating function.
(c) Using (b), find its first six momen
Homework Problem Set 5
(4.2.2) 1. A radioactive material emits -particles for which the density function for the time
between particle emissions is given by
f (t) = .1e.1t .
Find the probability that a particle is emitted in the first 10 seconds, given th
Homework Problem Set 10
(7.1.10) 1. (Levy1 ) Assume that n is an integer, not prime. Show that you can find two distributions a and b on the nonnegative integers such that the convolution of a and b is the
equiprobable distribution on the set 0, 1, 2, . .
Homework Problem Set 11
(8.1.5) 1. Let X be a random variable with E(X) = 0 and V (X) = 1. What integer value k will
assure us that P (|X| k) .01?
(Rice 5.7) 2. Show that if Xn c in probability and if g is a continuous function, then g(Xn ) g(c)
in probab
4.19)
Px(b) = cfw_
0 | b < 0
1/2 | 0 <= b < 1
1/10 | 1 <= b < 2
1/5 | 2 <= b < 3
1/10 | 3 <= b < 3.5
1/10 | b >= 3.5
4.20)
outcomes = cfw_
$0 | (20/38)^3
$1 | (18/38) + 2(20/38)(18/38)
$2 | (20/38)(18/38)^2
b) No because if you play two more times aft
Bernoulli trials
An experiment, or trial, whose outcome can be classified as either a success or failure is
performed.
X=
1 when the outcome is a success
0 when outcome is a failure
If p is the probability of a success then the pmf is,
p(0) =P(X=0) =1-p p