Consider a single tossing of x unbiased coins: a) Find the probability pX(x) of obtaining all heads; [2]

b) If x is regarded as a continuous variable in the range 0 ≤ x < ∞, show

that pX(x), regarded as a function of x and suitably scaled, determines

a probability distribution; [5]

c) Obtain a pdf of this distribution and calculate its cumulant generating

function; [5]

d) Hence or otherwise, show that the mean and variance are 1

ln 2 and

1

(ln 2)2 , respectively. [8]

e) Verify that the coefficient skewness for this distribution is γ3 = 2. [5]