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Discrete random variables and probability distributions 2.1. Suppose that a pair of fair dice are to be tossed, and let the random variable X denote the sum of the points. Obtain the probability distribution for X.
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The sample points for tosses of a pair of dice are given in Fig. 1-9, page 14. The random variable X is the sum of the coordinates for each point. Thus for (3, 2) we have X 5. Using the fact that all 36 sample points are equally probable, so that each sample point has probability 1 > 36, we obtain Table 2-4. For example, corresponding to X 5, we have the sample points (1, 4), (2, 3), (3, 2), (4, 1), so that the associated probability is 4 > 36.
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Table 2-4 x 2 3 2 > 36 4 3 > 36 5 4 > 36 6 5 > 36 7 6 > 36 8 5 > 36 9 4 > 36 10 3 > 36 11 2 > 36 12 1 > 36
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f (x) 1 > 36
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CHAPTER 2 Random Variables and Probability Distributions
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2.2. Find the probability distribution of boys and girls in families with 3 children, assuming equal probabilities for boys and girls.
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Problem 1.37 treated the case of n mutually independent trials, where each trial had just two possible outcomes, A and A , with respective probabilities p and q 1 p. It was found that the probability of getting exactly x A s in the n trials is nCx px qn x. This result applies to the present problem, under the assumption that successive births (the trials ) are independent as far as the sex of the child is concerned. Thus, with A being the event a boy, n 3, 1 and p q 2, we have
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P(exactly x boys)
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1 x 1 3 R Q R 2 2
3Cx Q
1 3 R 2
where the random variable X represents the number of boys in the family. (Note that X is defined on the sample space of 3 trials.) The probability function for X, f (x) is displayed in Table 2-5. Table 2-5 x f(x) 0 1> 8 1 3> 8 2 3> 8 3 1>8
3Cx Q
1 3 R 2
Discrete distribution functions 2.3. (a) Find the distribution function F(x) for the random variable X of Problem 2.1, and (b) graph this distribution function.
(a) We have F(x) P(X x) gu
f (u). Then from the results of Problem 2.1, we find 0 1>36 3>36 g6>36 ( 35>36 1 ` 2 3 4 11 12 x x x x ( x x 2 3 4 5 12 `
F(x)
(b) See Fig. 2-6.
Fig. 2-6
2.4. (a) Find the distribution function F(x) for the random variable X of Problem 2.2, and (b) graph this distribution function.
CHAPTER 2 Random Variables and Probability Distributions
(a) Using Table 2-5 from Problem 2.2, we obtain
F(x)
(b) The graph of the distribution function of (a) is shown in Fig. 2-7.
0 1>8 e 1>2 7>8 1
` 0 1 2 3
x x x x x
0 1 2 3 `
Fig. 2-7
Continuous random variables and probability distributions 2.5. A random variable X has the density function f(x) c > (x2 1), where the constant c. (b) Find the probability that X2 lies between 1 > 3 and 1.
` . (a) Find the value of
(a) We must have 3
f (x) dx
1, i.e.,
so that c (b) If 1 3 X2 1> . 1, then either
! 3>3
c dx
c tan
23 3
1 or
23 . Thus the required probability is 3
p R 2
1 p3
dx x2 1
1 1 p3
dx x2 1
!3>3
2 1 p3
dx 1 tan 1 6
2 p Btan 2 p p 4 1
x2 !3>3
1(1)
2.6. Find the distribution function corresponding to the density function of Problem 2.5.
F(x)
f (u) du
1 x p3 tan
u2 `)]
1 p [tan 1 2
1 p tan
1 p Btan
x 1 1 p Btan u Z ` R 1x
p 6
23 R 3
p R 2
CHAPTER 2 Random Variables and Probability Distributions
2.7. The distribution function for a random variable X is F(x) e 1 0 e
3 X 4.
Find (a) the density function, (b) the probability that X
2, and (c) the probability that
f (x)
d F(x) dx
2e 0 e
0 0 e
Another method
32 2e 1 P(X
2u du
2u P
By definition, P(X
F(2)
e 4. Hence, 2)
1 f (u) du
(1 3 1
e 4)
P( 3
0 du
Another method
2u 4 0
30 2e
2u du
P( 3
P(X F(4) (1
F( 3) e 8) (0) 1 e
Joint distributions and independent variables 2.8. The joint probability function of two discrete random variables X and Y is given by f(x, y) c(2x x and y can assume all integers such that 0 x 2, 0 y 3, and f(x, y) 0 otherwise.
(a) Find the value of the constant c. (b) Find P(X 2, Y 1). (c) Find P(X 1, Y 2).