4 16 marks A coin is tossed twice a Using H to denote heads

4. [16 marks ] A coin is tossed twice.

(a) Using H to denote heads and T to denote tails, list all elements of the sample space S .

(b) If the coin is unbalanced and a head has a 40% chance of occurring, determine the probability that atleast one head occurs.

(c) Let the random variable X denote the number of heads on the rst toss and let the random variable Y

denote the total number of heads on the 2 tosses. Again assuming the coin is unbalanced and a head has

a 40% chance of occurring, nd the joint probability distribution of X and Y . Show all work.

X = 0 X = 1

Y = 0

Y = 1

Y = 2

(d) Determine the marginal distribution of X and the marginal distribution of Y: State each in table form.

(e) What do the marginal distributions for X and Y represent?

(f) Calculate E (X ) and E (Y ).

(g) Determine the conditional probability distribution of X given Y = 1: Explain in words what this distribution represents.

(h) Use your results in (g) to compute E (XjY = 1) : Interpret this value.

(i) Are X and Y statistically independent variables? Explain using calculations to support your answer.

Solution

A coin is tossed twice.

(a) Using H to denote heads and T to denote tails, list all elements of the sample space S .

[ (H,H) , (H,T), (T,H) , (T,T) ]

(b) If the coin is unbalanced and a head has a 40% chance of occurring, determine the probability that atleast one head occurs.

P = 2*(2/5)*(3/5) + 2/5*2/5

P = 0.48+0.16 = 0.64