Schaum's Outline of Probability and Statistics, 3rd Ed. by John Schiller, R. Alu Srinivasan, Murray Spiegel

By John Schiller, R. Alu Srinivasan, Murray Spiegel

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

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13. 11. 11 it is seen that the joint density function for X and Y can be written as the product of a function of x alone and a function of y alone. In fact, f (x, y) ϭ f1(x)f2( y), where f1 (x) ϭ e c1x 0 0 Ͻ x Ͻ4 otherwise f2(y) ϭ e c2 y 0 1 Ͻ y Ͻ 5 otherwise and c1c2 ϭ c ϭ 1 > 96. It follows that X and Y are independent, so that their joint distribution function is given by F(x, y) ϭ F1(x)F2( y). 12, and Fig. 2-9 shows the resulting piecewise definition of F(x, y). 14. 11 find P(X ϩ Y Ͻ 3). Fig.

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 ϭ 5, we have the sample points (1, 4), (2, 3), (3, 2), (4, 1), so that the associated probability is 4 > 36. 2. Find the probability distribution of boys and girls in families with 3 children, assuming equal probabilities for boys and girls. 37 treated the case of n mutually independent trials, where each trial had just two possible outcomes, A and AЈ, with respective probabilities p and q ϭ 1 Ϫ p.

Suppose also that these values are assumed with probabilities given by P(X ϭ xk) ϭ f(xk) k ϭ 1, 2, . . (1) It is convenient to introduce the probability function, also referred to as probability distribution, given by P(X ϭ x) ϭ f(x) For x ϭ xk, this reduces to (1) while for other values of x, f(x) ϭ 0. In general, f (x) is a probability function if 1. f (x) Ն 0 2. a f (x) ϭ 1 x where the sum in 2 is taken over all possible values of x. 1. Assuming that the coin is fair, we have P(HH ) ϭ 1 4 P(HT ) ϭ 1 4 P(TH ) ϭ 1 4 P(T T ) ϭ 1 4 Then P(X ϭ 0) ϭ P(T T) ϭ 1 4 P(X ϭ 1) ϭ P(HT < TH ) ϭ P(HT ) ϩ P(TH ) ϭ P(X ϭ 2) ϭ P(HH) ϭ 1 1 1 ϩ ϭ 4 4 2 1 4 The probability function is thus given by Table 2-2.

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