Description
(Problems 1-8)
#1. Consider the following joint probability distribution of X and Y
Y \ X 1 2 3 4
4 0 1/20 1/20 1/20
3 1/20 2/20 3/20 1/20
2 1/20 2/20 3/20 1/20
1 1/20 1/20 1/20 0
a) Find marginal distributions
P(x)
and
P(y).
b) If
Z X 2Y
, find
P(z).
c) Using b), find
E(Z) .
d) Find
E(X) and
E(Y)
then find
E(Z)
using the expectation of X and expectation of Y
e) Are X and Y independent?
#2. Let joint Probability distribution function of random variables X and Y be
.
( , ) (0, 0)
( , ) (2, 0)
( , ) (1, 1)
0
1/ 3
1/ 3
1/ 3
( , )
otherwise
if x y
if x y
if x y
P x y
Are X and Y independent?
#3. Each morning John eats some eggs. On any given morning, the number of eggs he eats is equally
likely to 1, 2, 3, 4, or 5 independent of what he has done in the past. Let X be the number of eggs
that John eats in 10 days. Find the mean and the variance of X.
#4. The time till failure of an electronic component has an Exponential distribution and it is known
that 10% of components have failed by 1000 hours.
(a) What is the probability that a component is still working after 5000 hours?
(b) Find the mean and standard deviation of the time till failure.
#5. Let X be a continuous random variable with PDF,
otherwise
ax x
f x X
:
: 0 2.
0
( )
2
a) Find
a .
b) Find variance of X.
c) Find Cumulative Distribution Function (CDF) of X.
#6. Let X be a continuous random variable with PDF,
otherwise
ke x
f x
x
X
:
: 0.
0
( )
2
a) Find
k .
b) Find mean of X.
c) Find variance of X.
d) Find CDF of X.
#7. Let X be a discrete random variable with probability distribution (probability mass function),
x
P(x) c (1/3) , x 0, 1, 2,
a) Find
c
such that
P(x)
is a legitimate PMF.
b) Find Cumulative Distribution Function (cdf) of X, F(x) , x{0, 1, 2, }
#8. Suppose X has probability density function
otherwise
x x
f x X
:
: 0 1.
0
4
( )
3
a) Find cdf of X.
b) Using the answer in art a), find
P(X 1/ 2).
c) Using the answer in art a), find
P(1/3 X 2/3).
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Suggested Problems
Chapter-3: 60-66, Chapter-5: 2,4,5,6,8,9,10,11.
