Exercise 3.9

Answers

(a)

F (x) = p F_{1} (x) + (1 - p) F_{2} (x)

Let $x_{1} < x_{2}$ . Then

F (x_{1}) = p F_{1} (x_{1}) + (1 - p) F_{2} (x_{1}) < p F_{1} (x_{2}) + (1 - p) F_{2} (x_{2}) = F (x_{2}) .

Since $F (x)$ is a weighted sum of right continuous functions, it is itself a right continuous function.

\lim_{x \to \infty} F (x) = p \lim_{x \to \infty} F_{1} (x) + (1 - p) \lim_{x \to \infty} F_{2} (x) = p + 1 - p = 1 .

Similarly,

\lim_{x \to - \infty} F (x) = 0 .

(b)

Let

X

be an r.v. created as described. Let

H

be the event that coin lands heads, and

T

be the event that the coin lands tails.

Then, $F (X = k) = P (H) F_{1} (k) + P (T) F_{2} (k) = p F_{1} (k) + (1 - p) F_{2} (k) .$

Note that this is the same CDF as in part $a$ .

fifthist

2021-12-05 00:00