Calculate the mean, variance & standard deviation of the number of heads in a simultaneous toss of three coins.
SOLUTION:
Let X denotes the number of heads in a simultaneous toss of three coins. Then, X can take values 0,1,2,3.
NOW,
P(X=0)=P(TTT)=1/8
P(X=1)=P(HTT or TTH or THT)=3/8
P(X=2)=P(HHT or THH or HTH)=3/8 And,
P(X=3)=P(HHH)=1/8
Thus, the probability distribution of X is given by
X: 0 1 2 3
P(X):1/8 3/8 3/8 1/8
Computation of mean and variance
x_{i}

0

1

2

3


p_{i}

1/8

3/8

3/8

1/8


P_{i} x_{i} : 0 3/8 6/8 3/8 ∑pixi=3/2
P_{i} x_{i}²: 0 3/8 12/8 9/8 ∑pixi=3
WE HAVE,

X= MEAN = ∑ P_{i} x_{i} = 3/2
Var(X)= ∑ P_{i} x_{i} ² – (∑pi xi)² = 3 – (3/2)² = ¾
Standard deviation = √3/2 = 0.87