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

             P_i x_i :  0          3/8          6/8        3/8          ∑pixi=3/2 

           P_i x_i²:       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