A discrete random variable $X$ has the following probability distribution:
| $X$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
|---|---|---|---|---|---|---|---|---|---|
| $P \left(X\right)$ | $a$ | $3a$ | $5a$ | $7a$ | $9a$ | $11a$ | $13a$ | $15a$ | $17a$ |
- Find the value of $a$
- Find $P \left(x < 3\right)$
- Find $P \left(3 < x < 7\right)$
- $\sum \limits_{i=0}^{8} p_i = 1 $
Here $\;$ $p_0 = a, \; p_1 = 3a, \; p_2 = 5a, \; \cdots \; p_8 = 17a$
Now, $\;$ $a + 3a + 5a + 7a + 9a + 11a + 13a + 15a + 17a = 81a$
$\therefore \;$ $81 a = 1$ $\implies$ $a = \dfrac{1}{81}$
- $P \left(x < 3\right) = P \left(0\right) + P \left(1\right) + P \left(2\right)$
$\begin{aligned} \therefore \; P \left(x < 3\right) & = a + 3a + 5a \\\\ & = 9a \\\\ & = 9 \times \dfrac{1}{81} = \dfrac{1}{9} \end{aligned}$
- $P \left(3 < x < 7\right) = P \left(4\right) + P \left(5\right) + P \left(6\right)$
$\begin{aligned} \therefore \; P \left(3 < x < 7\right) & = 9a + 11a + 13a \\\\ & = 33a \\\\ & = 33 \times \dfrac{1}{81} = \dfrac{11}{27} \end{aligned}$