Question

If a car agency sells 50% of its inventory of a certain foreign equipped cars with air bags. Find out Probability mass function (P.M.F) & cumulative distribution function (C.D.F) of the next 4 cars to be sold

Answer 1

Let X be number of cars with airbags among the next 4 cars sold by the agency.

Therefore

$X\backsim Binomial (n=4,p=0.4)\\$

Probability mass function f(x)

$f(x)=\dbinom{4}{x}0.5^x(1-0.5)^{4-x}=\\ \dbinom{4}{x}0.5^x(1-0.5)^{4-x}=\dbinom{4}{x}0.5^4=\dfrac{1}{16}\dbinom{4}{x}\\ x=0,1,2,3,4$

Cumulative distribution function

$F(x)=\displaystyle\sum_{t\leq{x}}f(t)\\ f(0)=\dfrac{1}{16}\dbinom{4}{0}=\dfrac{1}{16}\\ f(1)=\dfrac{1}{16}\dbinom{4}{1}=\dfrac{1}{4}\\ f(2)=\dfrac{1}{16}\dbinom{4}{2}=\dfrac{3}{8}\\ f(3)=\dfrac{1}{16}\dbinom{4}{3}=\dfrac{1}{4}\\ f(4)=\dfrac{1}{16}\dbinom{4}{4}=\dfrac{1}{16}\\ F(0)=\dfrac{1}{16}\\ F(1)=f(0)+f(1)=\dfrac{5}{16}\\ F(2)=f(0)+f(1)+f(2)=\dfrac{11}{16}\\ F(3)=f(0)+f(1)+f(2)+f(3)=\dfrac{15}{16}\\ F(4)=f(0)+f(1)+f(2)+f(3)+f(4)=1\\ F(X)=\begin{cases} \dfrac{1}{16} &\text{for } 0\leq{x}\gt1\\ \\ \dfrac{5}{16} &\text{for } 1\leq{x}\gt2\\ \\ \dfrac{5}{16} &\text{for } 2\leq{x}\gt3\\ \\ \dfrac{15}{16} &\text{for } 3\leq{x}\gt4\\ \\ 1 &\text{for }x\geq4 \end{cases}$