Engineering Probability & Statistics
Sharif University of Technology
Hamid R. Rabiee & S. Abbas Hosseini
October 17, 2014
CE 181
Date Due: Aban 3rd , 1393
Homework 3
Problems
1. Four buses carrying 148 students from the same school arrive at a football stadium. The buses
carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let
X denote the number of students that were on the bus carrying the randomly selected student.
One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on her
bus.
(a) Which of E[X] or E[Y ] do you think is larger? Why?
(b) Compute E[X] and E[Y ].
2. An urn contains m + n balls, numbered 1, 2, . . . , m + n. A set of size n is drawn. If we let X
denote the number of balls drawn having numbers that exceed each of the numbers of those
remaining, compute the probability mass function of X.
3. A random variable X is said to be a Poisson random variable with parameter λ > 0 if
P {X = i} = e−λ
λi
,
i!
i = 0, 1, . . .
Let X be a Poisson random variable with parameter λ. Show that
1
1 + e−2λ
P {X is even} =
2
4. Suppose that X is a random variable with the following probability density function:
1
2 (x + 1) −1 ≤ x ≤ 1
f (x) =
0
o.w.
Compute the pdf of Y = 2X + 3.
5. The nonnegative continuous random variable X is said to have an exponential distribution with
parameter λ if its density function is
f (x) = λe−λx , x ≥ 0
(a) Let Y1 , Y2 be independent exponential random variables with parameters λ1 , λ2 respectively.
Find the value of P {Y1 < Y2 }.
(b) Let Y1 , . . . , Yn be independent exponential random variables with parameters λ1 , . . . , λn
respectively. Find the value of P {Yj = mini Yi } for 1 ≤ j ≤ n.
6. Let X be a continuous random variable with probabiliry distribution function F (X) and Let
Y = F (X). prove that Y has a uniform distribution over (0, 1).
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7. (a) Markov’s Inequality. Prove that if X is a nonnegative random variable, then for any
c > 0,
E[X]
P {X ≥ c} ≤
c
(b) Boole’s Inequality. Let A1 , . . . , An be n different events. Using the preceding part, prove
that
!
n
n
[
X
P
Ai ≤
P (Ai )
i=1
i=1
8. Suppose that there are m different types of coupons, and that each time one obtains a coupon,
it is equally likely to be any of these types. Find the expected value of the number of distinct
types in a collection of n coupons.
Hint: Break the desired random variable into indicator random variables.
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