Question

We have two coins whose heads are marked 2 and tails marked 1. One is a fair coin and the other is a biased coin whose probabilities of 'Head' are 1/2 and 1/4 respectively.Suppose we toss the two coins simultaneously. Let S and P be the sum and product of all the outcome numbers on the coins, respectively. 1. Compute the mean and variance of S. Calculate up to 3 decimal places (round the number at 4th place) if necessary. 2. Compute the mean of P

Answer 1

Let X1 shows the outcome of fair coin and X2 shows the outcome of biased coin.

P(X1=2) = 1/2 = 0.5, P(X1=1) = 1/2 =0.5

P(X2=2) = 1/4 = 0.25, P(X2=1) = 1 - (1/4) = 3/4 =0.75

Since both coins are independent so

P(X1=x1 and X2=x2) = P(X1=x1)P(X2=x2)

1:

Following table shows the calculations for S and corresponding probabilities:

X1	X2	P(X1=x1)	P(X2=x2)	S=X1+X2	P(S=s)=P(X1=x1)P(X2=x2)
2	2	0.5	0.25	4	0.125
2	1	0.5	0.75	3	0.375
1	2	0.5	0.25	3	0.125
1	1	0.5	0.75	2	0.375
Total					1

Now we need to sum the probabilites of S for common values of S. Following table show the pdf of S and calculations for mean and variance:

S	P(S=s)	s*P(S=s)	s^2*P(S=s)
4	0.125	0.5	2
3	0.5	1.5	4.5
2	0.375	0.75	1.5
Total	1	2.75	8

The mean of S is:

The variance of S is:

2:

Following table shows the calculations for P and corresponding probabilities:

X1	X2	P(X1=x1)	P(X2=x2)	P=X1*X2	P(P=p)=P(X1=x1)P(X2=x2)
2	2	0.5	0.25	4	0.125
2	1	0.5	0.75	2	0.375
1	2	0.5	0.25	2	0.125
1	1	0.5	0.75	1	0.375
Total					1

Now we need to sum the probabilites of P for common values of P. Following table show the pdf of P and calculations for mean and variance:

P	P(P=p)	p*P(P=p)	p^2*P(P=p)
4	0.125	0.5	2
2	0.5	1	2
1	0.375	0.375	0.375
Total	1	1.875	4.375

The mean of S is:

The variance of P is: