Geometric distribution

Exercise11

1.

(a)

For $X \sim G_{e}(p)$, show

$$E(X) = \frac{q}{p}, \ V(X) = \frac{q}{p^2}$$

Here,，$q = 1 - p$.

(b)

There is a team with a winning percentage of $30\%$. On average, find out how many games you need to play to win for the first time.

(c)

There are 10 lots, 4 of which are winning and the remaining 6 are out. Now draw 3 lots and find the probability of winning 2 of them.．

Answer

1.

(a)

$$\eta(t) = E(t^X) = \sum_{k=0}^{\infty}t^{k}P(X=k)$$

$$= \sum_{k=0}^{\infty}t^{k}(1-p)^{k}p$$

Then differentiate both sides with respect to $t$.

$$\eta'(t) = \sum_{k=1}^{\infty}kt^{k-1}(1-p)^{k}p$$

Set $t=0$. Then

$$\eta'(0) = E(X) = \sum_{k=1}^{\infty}k(1-p)^{k}p = p\sum_{k=1}^{\infty}kq^{k}$$

Let $S = \sum_{k=1}^{\infty}kq^{k}$. Then find $S - qS = S(1-q) = Sp$

$$Sp = S - qS = \sum_{k=1}^{\infty}kq^{k} - \sum_{k=1}^{\infty}kq^{k+1}$$

$$= \sum_{k=1}^{\infty}q^{k} = \frac{q}{1-q} = \frac{q}{p}$$

Thus, $S = \frac{q}{p^{2}}$ and

$$E(X) = p \frac{q}{p^{2}} = \frac{q}{p}$$

(b) Let $X$ be the numbr of games to play to win the first time. Then $X \sim G_{e}(0.3)$．Thus the number of games to play to win the first time is $E(X) + 1$

$$E(X) + 1 = \frac{q}{p} + 1 = \frac{0.7}{0.3} + 1 = 3.33 .$$

(c) If $X$ is the number of winning lottery, $X \sim H_{g}(10,4,3)$. Therefore, if you draw 3 lots, the probability of winning 2 of them is

$$P_{r}(X = 2) = \frac{{4 \choose 2}{6 \choose 1}}{{10 \choose 3}} = \frac{3}{10}$$