Question

if x is a random variable showing the number of boys in families with 4 children

a. construct a table showing the probability distribution(probability distribution table)

b. find the E(X)

c. Find the Var(X)

Answer 1

Since the probability of having a boy is not given, we assume it to be $p=\frac{1}{2}$

$P\left( X=0 \right) =\left( \frac{1}{2} \right) ^4=0.0625\\P\left( X=1 \right) =C_{4}^{1}\left( \frac{1}{2} \right) ^4=0.25\\P\left( X=2 \right) =C_{4}^{2}\left( \frac{1}{2} \right) ^4=0.375\\P\left( X=3 \right) =C_{4}^{3}\left( \frac{1}{2} \right) ^4=0.25\\P\left( X=4 \right) =\left( \frac{1}{2} \right) ^4=0.0625$

$EX=1\cdot 0.25+2\cdot 0.375+3\cdot 0.25+4\cdot 0.0625=2\\EX^2=1^2\cdot 0.25+2^2\cdot 0.375+3^2\cdot 0.25+4^2\cdot 0.0625=5\\Var\left( X \right) =EX^2-\left( EX \right) ^2=5-2^2=1\\$