Question

Three fair coins are tossed. Let X represent the number of tails.

           i.         Find the probability distribution of X.                                               (3 marks)

       ii.        Find E(X). Show that this is equivalent to  .                               (2 marks)

     iii.       Find Var(X). Show that this is equivalent to    .                        (3 marks)

     iv.       Find .                                                                                    (2 marks)

                                           

Answer 1

i. Probability distribution of X

X = number of tails

If three coins are tossed then following events will occur:

S = [HHT, HTH, HHH, THH, HTT, TTH, THT, TTT]

Total no. of outcomes =2³= 8

$P(X=0) = P(TTT) = \frac{1}{8}$

$P(X=1) = P(HTT,THT,TTH) = \frac{3}{8}$

$P(X=2) = P(HHT,HTH,THH) = \frac{3}{8}$

$P(X=3) = P(HHH) = \frac{1}{8}$

Following table shows the probability distribution of X:

ii. Find E(X)

$E(X) = (0\times\frac{1}{8})+(1\times\frac{3}{8})+(2\times\frac{3}{8})+(3\times\frac{1}{8})$

$E(X)= 0+0.375+0.75+0.375=1.5$

iii. Find Var(X)

$Var(X)=\Sigma X^{2}P(X)-E(X)^{2}$

$Var(X)=3-(1.5)^{2}=0.75$

iv. Question is not given.