3.1.8 problem 9

(a)

$F\left(x\right)=p{F}_1\left(x\right)+(1-p)F_2\left(x\right)$.

Let $x_1<x_2$. Then

$$F\left(x_1\right)=p{F}_1\left(x_1\right)+(1-p)F_2\left(x_1\right)<p{F}_1\left(x_2\right)+(1-p)F_2\left(x_2\right)=F\left(x_2\right).$$

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

$$\underset{x\to \infty }{\lim }F\left(x\right)=p\underset{x\to \infty }{\lim }{F}_1\left(x\right)+(1-p)\underset{x\to \infty }{\lim }{F}_2\left(x\right)=p+1-p=1.$$

Similarly,

$$\underset{x\to -\infty }{\lim }F\left(x\right)=0.$$

(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, $F(X=k)=P\left(H\right)F_1\left(k\right)+P\left(T\right)F_2\left(k\right)=p{F}_1\left(k\right)+(1-p)F_2\left(k\right).$

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