1. Given here is the joint probability mass function
𝑝𝑝𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦)
y
0
1
2
Total
0
0.38
0.14
0.24
0.76
x
1
0.17
0.02
0.05
0.24
Total
0.55
0.16
0.29
1
a)
b)
c)
d)
e)
f)
Find 𝐹𝐹𝑋𝑋𝑋𝑋 (1, 2). What is the interpretation of this value?
Find marginal distributions of X and Y.
Are X and Y independent?
Find conditional distributions of X|Y=2 and Y|X=1
Find E[X|Y=2] and Var[X|Y=2], and E[Y|X=1] and Var[Y|X=1]
Find covariance and correlation coefficient of X and Y
a)
b)
c)
d)
e)
f)
g)
Find the probabilities 𝑝𝑝𝑋𝑋𝑋𝑋 (3,1) and 𝑃𝑃[𝑋𝑋 < 3, 𝑌𝑌 > 1].
Find the conditional probabilities 𝑃𝑃[𝑋𝑋 = 3 | 𝑌𝑌 = 1] and 𝑃𝑃[𝑌𝑌 = 2 | 𝑋𝑋 < 3].
Find the joint cdf of X anf Y, 𝐹𝐹𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦).
Find the marginal probability mass functions of X and Y
Are X and Y independent? Explain.
Compute 𝐸𝐸[𝑋𝑋 | 𝑌𝑌 = 1] and 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 | 𝑌𝑌 = 1].
Compute the 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌) and the correlation coefficient 𝜌𝜌𝑋𝑋𝑋𝑋 .
2. Let the joint probability mass function of X and Y be
𝑥𝑥𝑦𝑦 2
𝑝𝑝𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦) = � 30 𝑥𝑥 = 1,2,3, 𝑦𝑦 = 1,2
0
𝑜𝑜. 𝑤𝑤.
[Hint: you can start by presenting 𝑝𝑝𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦) in form of a table]
3.
Let X and Y be discrete random variables such that
3𝑥𝑥 + 2𝑦𝑦
𝑥𝑥 = 1 or 2, 𝑦𝑦 = 𝑥𝑥 + 1 or 𝑥𝑥 + 2
𝑃𝑃[𝑋𝑋 = 𝑥𝑥, 𝑌𝑌 = 𝑦𝑦] = � 42
0
𝑜𝑜. 𝑤𝑤.
a) Find the marginal probability mass functions of X and Y
b) Compute 𝐸𝐸[𝑋𝑋 | 𝑌𝑌 = 3] and 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 | 𝑌𝑌 = 3].
c) Compute the 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌) and the correlation coefficient 𝜌𝜌𝑋𝑋𝑋𝑋 .
4. Let
2
2 )𝑒𝑒 −𝑥𝑥
, 0 ≤ 𝑥𝑥 < ∞, − 𝑥𝑥 ≤ 𝑦𝑦 < 𝑥𝑥
𝑓𝑓𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦) = �𝑐𝑐(𝑥𝑥 − 𝑦𝑦
0
𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
[Hint: you may choose to use Gamma function to compute some of the integrals]
a. Find c.
b. Find the probability 𝑃𝑃[𝑋𝑋 > 2𝑌𝑌].
c. Find 𝐸𝐸[𝑋𝑋𝑋𝑋].
d. Find the joint c.d.f. of X and Y, 𝐹𝐹𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦).
e. Find the marginal densities 𝑓𝑓𝑋𝑋 (𝑥𝑥) and 𝑓𝑓𝑌𝑌 (𝑦𝑦).
f. Are X and Y independent? Explain.
g. Find the conditional densities 𝑓𝑓𝑋𝑋|𝑌𝑌=𝑦𝑦 (𝑥𝑥) and 𝑓𝑓𝑌𝑌|𝑋𝑋=𝑥𝑥 (𝑦𝑦).
h. Find the probability 𝑃𝑃[𝑌𝑌 > 1/2|𝑋𝑋 = 1].
i. Find 𝐸𝐸[𝑌𝑌|𝑋𝑋 = 1].
j. Find 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌) and the correlation coefficient 𝜌𝜌𝑋𝑋𝑋𝑋 .
5. The joint density function of X and Y is given by
a.
b.
c.
d.
e.
f.
g.
h.
i.
𝑓𝑓𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦) = �𝑐𝑐𝑐𝑐𝑦𝑦
0
2
𝑥𝑥 − 1 ≤ 𝑦𝑦 ≤ 1 − 𝑥𝑥,
0 ≤ 𝑥𝑥 ≤ 1
elsewhere
Find 𝑐𝑐.
Find 𝐹𝐹𝑋𝑋𝑋𝑋 (1/2, 1/2).
Find 𝐹𝐹𝑋𝑋𝑋𝑋 (1/2, 2).
Find P(X > Y).
Find marginal densities of X and Y.
Find conditional densities of 𝑋𝑋|𝑌𝑌 = 𝑦𝑦 and 𝑌𝑌|𝑋𝑋 = 𝑥𝑥.
Find 𝐸𝐸[𝑋𝑋|𝑌𝑌 = 𝑦𝑦] and 𝐸𝐸[𝑌𝑌|𝑋𝑋 = 𝑥𝑥].
Are X and Y independent?
Find covariance and correlation coefficient of X and Y
6. The amount of time, T, between the occurrence and the reporting of an accident has pdf
8𝑡𝑡 − 𝑡𝑡 2
0 < 𝑡𝑡 < 6
𝑓𝑓𝑇𝑇 (𝑡𝑡) = � 72
0
𝑜𝑜. 𝑤𝑤.
Given that T=t, the amount of time between the reporting of the accident and the payment by the insurance
company is uniformly distributed on [2+t, 10].
Compute the probability that the amount of time between the occurrence of the accident and the payment by
the insurance company is less than 4.
7. Assume that X, Y, and Z are random variables, with
E(X) = 2,
E(Y) = −1,
E(Z) = 4,
Var(X) = 4,
Var(Y) = 6,
Var(Z) = 8,
Cov(X, Y) = 1,
Cov(X, Z) = −1,
Cov(Y, Z) = 0.
Find E(3X + 4Y – 6Z) and V(3X + 4Y – 6Z).
8. Let X and Y be independent random variables with 𝜇𝜇𝑋𝑋 = 1, 𝜇𝜇𝑌𝑌 = −1, 𝜎𝜎𝑋𝑋2 = 0.5, and 𝜎𝜎𝑌𝑌2 = 2.
Compute 𝐸𝐸[(𝑋𝑋 + 1)2 (𝑌𝑌 − 1)2 ].
9. Assume that X~Uniform[0, t], where t is exponentially distributed with mean 𝛽𝛽. Find E[X] and Var[X].
10. The Weibull density function is given by
𝑦𝑦 𝑚𝑚
1
𝑚𝑚−1 − 𝛼𝛼
𝑒𝑒
𝑓𝑓𝑌𝑌 (𝑦𝑦) = �𝛼𝛼 𝑚𝑚𝑦𝑦
0
𝑦𝑦 > 0
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑒𝑒𝑒𝑒𝑒𝑒
where α and m are positive constants. Suppose Y has the Weibull density just given. Find the density function
of 𝑈𝑈 = 𝑌𝑌 𝑚𝑚 . [Hint: notice that y>0, and use the method of transformations (justify first!!!).]
11. Let X and Y be independent standard normal random variables and U = X and V =X + Y.
a. Derive the joint density of U and V.
b. Are U and V independent? Why or why not?
12. X is a random variable with probability density function
−2𝑥𝑥
𝑓𝑓𝑋𝑋 (𝑥𝑥) = �𝑒𝑒 4𝑥𝑥
2𝑒𝑒
Let 𝑇𝑇 = 𝑋𝑋 2 . Find the probability density function of T.
13. Random variables X and Y have joint density function
𝑥𝑥 ≥ 0
𝑥𝑥 < 0
2 0 ≤ 𝑥𝑥 ≤ 𝑦𝑦 ≤ 1
𝑓𝑓𝑋𝑋𝑋𝑋 (𝑥𝑥, 𝑦𝑦) = �
0
𝑜𝑜. 𝑤𝑤.
Let 𝑍𝑍 = 1/𝑌𝑌 . Find the probability density function of Z.
𝜋𝜋 𝜋𝜋
14. Suppose that X follows a uniform distribution on the interval �− , �. Find the cdf and density of
2 2
𝑈𝑈(𝑋𝑋) = 𝑡𝑡𝑡𝑡𝑡𝑡(𝑋𝑋).
15. If the radius of a circle is an exponential random variable, find the density function of the area.
16. Let X and Y be jointly continuous random variables.
a. Develop an expression for the joint density of X + Y and X − Y .
b. Develop an expression for the joint density of XY and Y/X.
c. Specialize the expressions from parts (a) and (b) to the case where X and Y are independent.
17. Let 𝑋𝑋1 , 𝑋𝑋2 , … , 𝑋𝑋𝑛𝑛 be independent identically distributed variables with common density function
−(𝑥𝑥−𝜃𝜃)
𝑥𝑥 > 𝜃𝜃
𝑓𝑓𝑋𝑋 (𝑥𝑥) = �𝑒𝑒
0
𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
for some positive constant 𝜃𝜃. Find
a. The density function for 𝑈𝑈 = 𝑋𝑋(1) = min{𝑋𝑋1 , 𝑋𝑋2 , … , 𝑋𝑋𝑛𝑛 }
b. 𝐸𝐸�𝑋𝑋(1) �
18. Let X and Y be identically distributed independent random variables such that the moment generating function
of X + Y is
𝑀𝑀𝑋𝑋+𝑌𝑌 (𝑡𝑡) = 0.09𝑒𝑒 −2𝑡𝑡 + 0.24𝑒𝑒 −𝑡𝑡 + 0.34 + 0.24𝑒𝑒 𝑡𝑡 + 0.09𝑒𝑒 2𝑡𝑡
Calculate P[X ≤0].
19. X and Y are independent random variables with common moment generating function
𝑡𝑡 2
𝑀𝑀(𝑡𝑡) = 𝑒𝑒 2
Let 𝑊𝑊 = 𝑋𝑋 + 𝑌𝑌 and 𝑍𝑍 = 𝑌𝑌 − 𝑋𝑋.
Determine the joint moment generating function, 𝑀𝑀𝑊𝑊𝑊𝑊 (𝑠𝑠, 𝑡𝑡), of W and Z.
20. Let L be the number of students who attend office hours on any given day. L follows Poisson distribution with
mean 5. Using Chebyshev’s inequality, find the upper bound of 𝑃𝑃[|𝐿𝐿 − 5| ≥ 10]
21. A random variable X that assumes only nonnegative values has 𝐸𝐸[𝑋𝑋] = 16 and 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] = 36. Provide the upper
bound of the probability 𝑃𝑃[𝑋𝑋 > 46].
Review all examples in Lecture Notes
Review every problem on HW1 – HW6
Review every problem on Exam 1
Review every problem on Exam 1 Practice Problems