# Question

With reference to Theorem 3.3, verify that

(a) P(X > xi) = 1- F(xi) for i = 1, 2, 3, . . . , n;

(b) P(X G xi) = 1- F(xi- 1) for i = 2, 3, . . . , n, and P(X ≥ x1) = 1.

Theorem 3.3

If the range of a random variable X consists of the values x1 < x2 < x3 < · · · < xn, then f(x1) = F(x1) and f(xi) = F(xi) – F(xi – 1) for i = 2, 3, . . . , n

(a) P(X > xi) = 1- F(xi) for i = 1, 2, 3, . . . , n;

(b) P(X G xi) = 1- F(xi- 1) for i = 2, 3, . . . , n, and P(X ≥ x1) = 1.

Theorem 3.3

If the range of a random variable X consists of the values x1 < x2 < x3 < · · · < xn, then f(x1) = F(x1) and f(xi) = F(xi) – F(xi – 1) for i = 2, 3, . . . , n

## Answer to relevant Questions

Find the distribution function of the random variable X of Exercise 3.17 and use it to reevaluate part (b). The p. d. f. of the random variable X is given by Find (a) The value of c; (b) P(X < 1/4 ) and P(X > 1). Find the distribution function of the random variable X whose probability density is given by Also sketch the graphs of the probability density and distribution functions. With reference to Figure 3.9, find expressions for the values of the distribution function of the mixed random variable X for (a) x ≤ 0; (b) 0< x< 0.5; (c) 0.5 F x< 1; (d) x ≥ 1. Figure 3.9 F(x, y) is the value of the joint distribution function of two discrete random variables X and Y at (x, y), show that (a) F(-∞,-∞) = 0; (b) F(q, q) = 1; (c) if a< b and c< d, then F(a, c) ≤ F(b, d).Post your question

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