Question

1. A discrete random variable Y has a pmf given by p(Y=y)=c(3/4)^X for y=0,1,2,....Find the values of the constant c and p(x<3)

Answer 1

$$p(Y=y)=c(\dfrac{3}{4})^y, y=0,1,2,...$$

$$\displaystyle\sum_{y=0}^{\infin}p(y)=\displaystyle\sum_{y=0}^{\infin}c(\dfrac{3}{4})^y=\dfrac{c}{1-\dfrac{3}{4}}=4c=1$$

$$c=\dfrac{1}{4}$$

$$p(Y<3)=P(Y=0)+P(Y=1)+P(Y=2)$$

$$=\dfrac{1}{4}(\dfrac{3}{4})^0+\dfrac{1}{4}(\dfrac{3}{4})^1+\dfrac{1}{4}(\dfrac{3}{4})^2=\dfrac{37}{64}$$