Question

Verify whether f(x) = (8/7) (1/2)^x

, x = 1,2,3 is a probability mass function. If it is,

determine the following probabilities.

(a) P(X ≤ 1)

(b)P(X > 1)

(c) P(2 < X < 6)

(d)P(X ≤ 1 or X > 1)

Answer 1

$$f(x)=\dfrac{8}{7}(\dfrac{1}{2})^x,x=1,2,3$$

$$\dfrac{8}{7}(\dfrac{1}{2})^x>0,x=1,2,3$$

$$\dfrac{8}{7}(\dfrac{1}{2})^1+\dfrac{8}{7}(\dfrac{1}{2})^2+\dfrac{8}{7}(\dfrac{1}{2})^3=1$$

$$P(X=1)=\dfrac{8}{7}(\dfrac{1}{2})$$

$$P(X=2)=\dfrac{8}{7}(\dfrac{1}{4})$$

$$P(X=3)=\dfrac{8}{7}(\dfrac{1}{8})$$

$f(x)$ is a probability mass function.

(a)

$$P(X\le1)=\dfrac{4}{7}$$

(b)

$$P(X>1)=1-\dfrac{4}{7}=\dfrac{3}{7}$$

(c)

$$P(2<X<6)=P(X=3)=\dfrac{1}{7}$$

(d)

$$P(X\le 1\ or\ X>1)=1$$