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Problem Set 4: HANU - Faculty of Information and Technology Probability & Statistics

This document contains 15 probability and statistics problems from a problem set for a class. The problems cover topics like: 1. Finding joint and marginal probability functions for discrete and continuous random variables. 2. Determining constants in probability density functions. 3. Calculating probabilities based on probability density and distribution functions. 4. Determining if random variables are independent based on their joint distributions. 5. Finding conditional probability and density functions. The problems reference exercises from a probability and statistics textbook and include calculating various probabilities, densities, constants, and determining independence between random variables.

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66 views4 pages

Problem Set 4: HANU - Faculty of Information and Technology Probability & Statistics

This document contains 15 probability and statistics problems from a problem set for a class. The problems cover topics like: 1. Finding joint and marginal probability functions for discrete and continuous random variables. 2. Determining constants in probability density functions. 3. Calculating probabilities based on probability density and distribution functions. 4. Determining if random variables are independent based on their joint distributions. 5. Finding conditional probability and density functions. The problems reference exercises from a probability and statistics textbook and include calculating various probabilities, densities, constants, and determining independence between random variables.

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Ngọc Trâm
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Problem Set 4

HANU - Faculty of Information and Technology


Probability & Statistics

Problem 1: [3, Exercise 2.54]


The joint probability function of two discrete random variables X and Y is given by
f (x, y) = cxy for x = 1, 2, 3 and y = 1, 2, 3, and equals zero otherwise. Find (a) the
constant c, (b) P (X = 2, Y = 3), (c) P(1 ≤ X ≤ 2, Y ≤ 2), (d) P(X ≥ 2), (e)
P(Y < 2), (f) P(X = 1), (g) P(Y = 3).

Problem 2: [3, Exercise 2.55]


Find the marginal probability functions of (a) X and (b) Y for the random variables
of Problem 1. (c) Determine whether X and Y are independent.

Problem 3: [3, Exercise 2.56]


Let X and Y be continuous random variables having joint density function

c(x2 + y 2 ) 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
f (x, y) =
0, otherwise
1 1 1 3 1
Determine (a) the constant c, (b) P(X < , Y > ), (c) P( < X < ), (d) P(Y < ),
2 2 4 4 2
(e) whether X and Y are independent.

Problem 4: [3, Exercise 2.57]


Find the marginal distribution functions (a) of X and (b) of Y for the density function
of Problem 3.

Problem 5: [3, Exercise 2.58]


Find the conditional probability function (a) of X given Y , (b) of Y given X, for the
distribution of Problem 1.

Problem 6: [3, Exercise 2.59]


Let 
x + y 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
f (x, y) =
0, otherwise
Find the conditional density function of (a) X given Y , (b) Y given X.

Problem 7: [3, Exercise 2.60]


Find the conditional density function of (a) X given Y , (b) Y given X, for the distri-
bution of Problem 3.

1
Problem 8: [3, Exercise 2.61]
Let
e−(x+y) x ≥ 0, y ≥ 0

f (x, y) =
0, otherwise
be the joint density function of X and Y . Find the conditional density function of (a)
X given Y , (b) Y given X.

Problem 9: [3, Exercise 2.81]


Suppose that f (x) = c/3x , x = 1, 2, ..., is the probability function for a random vari-
able X. (a) Determine c. (b) Find the distribution function. (c) Graph the probability
function and the distribution function. (d) Find P (2 ≤ X < 5). (e) Find P (X ≥ 3).

Problem 10: [3, Exercise 2.82]


Suppose that
cxe−2x

x≥0
f (x) =
0, otherwise
is the density function for a random variable X. (a) Determine c. (b) Find the dis-
tribution function. (c) Graph the density function and the distribution function. (d)
Find P (X ≥ 1). (e) Find P (2 ≤ X < 3).

Problem 11: [3, Exercise 2.83]


The probability function of a random variable X is given by


 2p x=1
p x=2

f (x) =

 4p x =3
0, otherwise

where p is a constant. Find (a) P (0 ≤ X < 3), (b) P (X > 1).

Problem 12: [3, Exercise 2.84]


(a) Prove that for a suitable constant c,

0 x≥0
F (x) = − 2
c(1 − e x) , x > 0
is the distribution function for a random variable X, and find this c. (b) Determine
P (1 < X < 2).

Problem 13: [3, Exercise 2.86]


Two independent random variables, X and Y , have respective density functions
c1 e−2x x > 0

f (x) =
0 x≤0

c2 ye−3y y > 0

g(y) =
0 y≤0
Find (a)c1 and c2 ,(b) P (X + Y > 1),(c) P (1 < X < 2, Y ≥ 1),(d)P (1 < X <
2),(e)P (Y ≥ 1).

2
Problem 14: [3, Exercise 2.88]
Let X and Y be random variables having joint density function

c(2x + y) 0 < x < 1, 0 < y < 2
f (x, y) =
0 otherwise

1 3
Find (a) the constant c, (b) P (X > , Y < ), (c) the (marginal) density function of
2 2
X, (d) the (marginal) density function of Y .

Problem 15: [3, Exercise 2.91]



1/y 0 < x < y, 0 < y < 1
f (x, y) =
0 otherwise
1
(a) Determine whether X and Y are independent, (b) Find P (X > ). (c) Find
2
1 1 1
P (X < , Y > ). (d) Find P (X + Y > ).
2 3 2

3
References
[1] Walpole, R. E., Myers, R. H., Myers, S. L. and Ye, K., Probability &
Statistics for Engineers & Scientists, 9th ed., MA, USA: Prentice-Hall, 2012.

[2] DeGroot, M. H. and Schervish, M. J., Probability and Statistics, 4th ed., MA,
USA: Pearson Education, Inc., 2012.

[3] Murray, R. S., John, J. S. and R, A. Srinivasan, Probability and Statistics,


3rd ed., USA: McGraw-Hill, 2009.

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