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**Additional resources for Schaum's Outline of Probability and Statistics, 3rd Ed. (Schaum's Outline Series)**

**Example text**

S S x Ϫ` x ` 3. , limϩF(x ϩ h) ϭ F(x) for all x]. hS0 Distribution Functions for Discrete Random Variables The distribution function for a discrete random variable X can be obtained from its probability function by noting that, for all x in (Ϫ ` , ` ), F(x) ϭ P(X Յ x) ϭ a f (u) (4) uՅx where the sum is taken over all values u taken on by X for which u Յ x. If X takes on only a finite number of values x1, x2, . . 2. (b) Obtain its graph. (a) The distribution function is F(x) ϭ 0 Ϫ` 0 1 2 1 1 d 43 4 ϽxϽ0 Յ xϽ1 ՅxϽ2 ՅxϽ` (5) 36 CHAPTER 2 Random Variables and Probability Distributions (b) The graph of F(x) is shown in Fig.

A box contains 2 red and 3 blue marbles. Find the probability that if two marbles are drawn at random (without replacement), (a) both are blue, (b) both are red, (c) one is red and one is blue. 53. Find the probability of drawing 3 aces at random from a deck of 52 ordinary cards if the cards are (a) replaced, (b) not replaced. 54. If at least one child in a family with 2 children is a boy, what is the probability that both children are boys? 55. Box I contains 3 red and 5 white balls, while Box II contains 4 red and 2 white balls.

5 (a) Find the constant c such that the function f (x) ϭ b cx2 0 0ϽxϽ3 otherwise is a density function, and (b) compute P(1 Ͻ X Ͻ 2). (a) Since f (x) satisfies Property 1 if c Ն 0, it must satisfy Property 2 in order to be a density function. Now ` 3 3 cx3 2 2 ϭ 9c 3Ϫ` f (x) dx ϭ 30 cx dx ϭ 3 0 and since this must equal 1, we have c ϭ 1 > 9. 2 2 1 x3 2 8 1 7 P(1 Ͻ X Ͻ 2) ϭ 3 x2 dx ϭ ϭ Ϫ ϭ 27 1 27 27 27 1 9 (b) In case f (x) is continuous, which we shall assume unless otherwise stated, the probability that X is equal to any particular value is zero.