Engineering Probability & Statistics
Sharif University of Technology
Hamid R. Rabiee & S. Abbas Hosseini
November 7, 2014
CE 181
Date Due: Azar 1st , 1393
Homework 5 (Chapter 5)
Problems
1. Suppose Y = X 2 − 2X and:
X
P (X)
-2
0.1
-1
0.2
0
0.4
1
0.2
2
0.1
Find E[Y ] and var[Y ].
2. If X is uniformly distributed on (0, 1) and Y = eX , find pdf and cdf of Y .
3. Suppose r.v. X has a standard Cauchy distribution with parameter π:
f (x) =
Show that Y =
1
X
1
π(1 + x2 )
has a standard Cauchy distribution too.
4. Standard score of r.v. X is defined as:
Z=
X −µ
σ
where µ is its mean and σ is its standard devation. Skewness of real valued r.v. X is the third
moment of the standard score of X.
skew(X) = E[Z 3 ]
It is a measure of the assymetry of the probability distribution of X. A positively skewed distribution tends to have a long tail to the right. Similarly, a negatively skewed distribution will
have a long tail to the left.
For a real valued r.v. X with a pdf f which is symmetric about a (f (a − t) = f (a + t), t ∈ R),
prove that skew(X) = 0.
(Hint: Show that E[X] = a)
1
5. Kurtosis of real valued r.v. X is the fourth moment of the standard score of X.
kurt(X) = E[Z 4 ]
It is a measure of the ”peakedness” of X. A distribution with a large kurtosis will have a sharp
peak and fat tails.
Suppose X has a uniform distribution in [a, b]. Compute kurt(X).
6. If E(X) = E(X 2 ) = 0, show that P (X = 0) = 1.
(Hint: Use Chebyshev inequality.)
7. Let X be a Poisson r.v., E[X] = 9 and Q = p{X ≥ 21}.
a. Use Markov inequality to find an upper bound for Q.
b. Use Chebyshev inequality to find an upper bound for Q.
8. Suppose r.v. X has a binomial distribution with parameters n and p. Show that its characteristic
function is equal to:
(pejω + q)n
where q is 1 − p.
9. Using the result of previous question, find mean and variance of X.
2