Question

There are 4 traffic lights along the road, each prohibits further movement of the car with a probability of 0.5. Find the

distribution of the number of the traffic lights that car passed to the first stop. What is the variance of the random

variable? 

Answer 1

Let $X=$ the number of the traffic lights that car passed to the first stop.

Then

$$P(X=0)=0.5(0.5)^0=0.5$$

$$P(X=1)=0.5(0.5)^1=0.25$$

$$P(X=2)=0.5(0.5)^2=0.125$$

$$P(X=3)=0.5(0.5)^3=0.0625$$

$$P(X=4)=0.5(0.5)^4=0.03125$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 & 4 \\ \hline p & 0.5 & 0.25 & 0.125 & 0.0625 & 0.03125 \\ \end{array}$$

$$E(X)=0(0.5)+1(0.25)+2(0.125)$$

$$+3(0.0625)+4(0.03125)=0.8125$$

$$E(X^2)=0^2(0.5)+1^2(0.25)+2^2(0.125)$$

$$+3^2(0.0625)+4^2(0.03125)=1.8125$$

$$Var(X)=E(X^2)-(E(X))^2$$

$$=1.8125-(0.8125)^2=1.15234375$$