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

Complicated Textbooks? ignored Lectures? no longer sufficient Time? thankfully for you, there is Schaum's Outlines. greater than forty million scholars have relied on Schaum's to assist them achieve the school room and on assessments. Schaum's is the most important to speedier studying and better grades in each topic. each one define provides all of the crucial path info in an easy-to-follow, topic-by-topic layout.

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

**Sample text**

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.