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2.52. Can the function F(x) be a distribution function Explain. 2.53. Let X be a random variable having density function f (x) Find (a) the value of the constant c, (b) P( 2
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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(l X 2, Y 2), (d) P(X 2), (e) P(Y 2), (f) P(X 1), (g) P(Y 3). 2.55. Find the marginal probability functions of (a) X and (b) Y for the random variables of Problem 2.54. (c) Determine whether X and Y are independent. 2.56. Let X and Y be continuous random variables having joint density function f (x, y) Determine (a) the constant c, (b) P(X independent. e
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1 2, Y
c(x 2 0
1 2 ),
y 2) (c) P ( 1 4
0 x 1, 0 otherwise X
3 4 ),
1 2 ),
(d) P(Y
(e) whether X and Y are
2.57. Find the marginal distribution functions (a) of X and (b) of Y for the density function of Problem 2.56.
Conditional distributions and density functions
2.58. Find the conditional probability function (a) of X given Y, (b) of Y given X, for the distribution of Problem 2.54. 2.59. Let f (x, y) e x 0 y x 1, 0 0 otherwise y 1
Find the conditional density function of (a) X given Y, (b) Y given X. 2.60. Find the conditional density of (a) X given Y, (b) Y given X, for the distribution of Problem 2.56. 2.61. Let f (x, y) e e 0
(x y)
x 0, 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.
CHAPTER 2 Random Variables and Probability Distributions
Change of variables
2.62. Let X have density function f (x) Find the density function of Y X2. e e 0
2.63. (a) If the density function of X is f (x) find the density function of X3. (b) Illustrate the result in part (a) by choosing f (x) and check the answer. 2.64. If X has density function f (x) 2(p)
1> 2e x2> 2,
2e 0
`, find the density function of Y
2.65. Verify that the integral of g1(u) in Method 1 of Problem 2.21 is equal to 1. 2.66. If the density of X is f (x) 1 > p(x2 1), ` x ` , find the density of Y tan
2.67. Complete the work needed to find g1(u) in Method 2 of Problem 2.21 and check your answer. 2.68. Let the density of X be e 1>2 0 1 x 1 otherwise
f (x) Find the density of (a) 3X 2, (b) X3 1.
2.69. Check by direct integration the joint density function found in Problem 2.22. 2.70. Let X and Y have joint density function f (x, y) If U X > Y, V X e e 0
(x y)
x 0, y otherwise
Y, find the joint density function of U and V. XY 2, (b) V X 2Y.
2.71. Use Problem 2.22 to find the density function of (a) U
2.72. Let X and Y be random variables having joint density function f (x, y) (2p) 1 e (x2 y2), ` x `, ` y ` . If R and are new random variables such that X R cos , Y R sin , show that the density function of R is g(r) e re 0
r2>2
CHAPTER 2 Random Variables and Probability Distributions
e 1 0 0 x 1, 0 otherwise y 1 XY.
2.73. Let
f (x, y)
be the joint density function of X and Y. Find the density function of Z
Convolutions
2.74. Let X and Y be identically distributed independent random variables with density function f (t) Find the density function of X e 1 0 0 t 1 otherwise
Y and check your answer.
2.75. Let X and Y be identically distributed independent random variables with density function f (t) Find the density function of X e e 0
t 0 otherwise
Y and check your answer. Z and then using convolutions to find the density
2.76. Work Problem 2.21 by first making the transformation 2Y function of U X Z.
2.77. If the independent random variables X1 and X2 are identically distributed with density function f (t) find the density function of X1 X2. e te 0
Applications to geometric probability
2.78. Two points are to be chosen at random on a line segment whose length is a three line segments thus formed will be the sides of a triangle. 0. Find the probability that the
2.79. It is known that a bus will arrive at random at a certain location sometime between 3:00 P.M. and 3:30 P.M. A man decides that he will go at random to this location between these two times and will wait at most 5 minutes for the bus. If he misses it, he will take the subway. What is the probability that he will take the subway 2.80. Two line segments, AB and CD, have lengths 8 and 6 units, respectively. Two points P and Q are to be chosen at random on AB and CD, respectively. Show that the probability that the area of a triangle will have height AP and that the base CQ will be greater than 12 square units is equal to (1 ln 2) > 2.
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