Question

Four coins are tossed. Let Z be the random variable representing the number of heads that occur. Find the values of the random variable Z. Compute for the mean, variance, standard deviation. Construct a Probability histogram. *

9 points

Answer 1

$Z\sim Bin\left( 4,0.5 \right) \\Z\in \left\{ 0,1,2,3,4 \right\} \\P\left( Z=0 \right) =0.5^4=0.0625\\P\left( Z=1 \right) =C_{4}^{1}\cdot 0.5\cdot 0.5^3=0.25\\P\left( Z=2 \right) =C_{4}^{2}\cdot 0.5^2\cdot 0.5^2=0.375\\P\left( Z=3 \right) =C_{4}^{3}\cdot 0.5\cdot 0.5^3=0.25\\P\left( Z=4 \right) =0.5^4=0.0625$

$EZ=0\cdot 0625+1\cdot 0.25+2\cdot 0.375+3\cdot 0.25+4\cdot 0.0625=2\\EZ^2=0^2\cdot 0625+1^2\cdot 0.25+2^2\cdot 0.375+3^2\cdot 0.25+4^2\cdot 0.0625=5\\DZ=EZ^2-\left( EZ \right) ^2=5-2^2=1\\\sigma Z=\sqrt{DZ}=1$