Question

There are 10 white and 5 black balls in the box. Four balls one after the other (returning ball to
the box) were taken from the box. Let X is the number of black ball selected.
a) Find a probability density function f(x) and a cumulative distributional function F(x);
b) What is the mean and the variance?
c) What is the probability that exactly 2 balls will be white?

Answer 1

$p=\frac{1}{3}, \, q=1-p,$ and $n=4$

(a). $f(x)=\binom{n}{x}×p^x×q^{n-x}, x=0,1,2,3,4,$

$F(x,i)=\sum_{i}^{x}\binom{n}{i}p^iq^{n-i}$

b. $E(x)=np=\frac{4}{3}=1\frac{1}{3}$

$Var(x)=npq=\frac{4}{3}×\frac{2}{3}=\frac{8}{9}$

c. from a, we have $f(2)=\binom{4}{2}×(\frac{1}{3})^2×(\frac{2}{3})^{4-2}$

$=6×\frac{1}{9}×\frac{4}{9}=\frac{24}{81}$