A six-sided die is rolled, and the number N on the uppermost face is recorded. Then a fair coin is tossed N times, and the total number Z of heads to appear is observed. Determine the mean and variance of Z by viewing Z as a random sum of N Bernoulli random variables. Determine the probability mass function of Z, and use it to find the mean and variance of Z.

1. Mean is 1.75

2. The variance is 1.6042

3.

The distribution function is:

Z           Z/K

0           21/128

1             5/16

2         33/128

3          1/6

4           29/384

5             1/48

6          1/384

Step-by-step explanation:

The mean of Z is given as:

E(Z) =Σ6, k=0 Kp (Z = k)

Σ6,k=0 K 1/6 Σ6, n=k (n k) (1/2)^n

=( 0(21/128) + 1(5/16) + 2( 33/128) + 3 (1/6) + 4 (29/384) + 5 (1/48) + 6 (1/384))

=7/4

=1.75

Thus, the mean Z is 1.75

The variance of Z is given as:

Var (Z) = E (Z^2) - (E (Z)) ^2

Therefore,

E(Z^2) = Σ 6, k=0 K^2P ( Z=K)

= ( 0(21/128 + 1(5/16) + 4(33/128) + 9(1/6) + 16(29/384) + 25(1/48) + 36(1/384))

=14/3

Var (Z) = 14/7 - (7/4)^2

= 14/7 - 49/16

=77/48

=1.6042

Thus, the variance is 1.6042

The probability of mass function is given as:

P(Z=k) = 1/6 Σ 6, n=k (n  k) (1/2)^n

The distribution function is

Z           Z/K

0           21/128

1             5/16

2         33/128

3          1/6

4           29/384

5             1/48

6          1/384

answer: c

because if you look at the chart you can see a pattern and compare it to the question he simplify out your

but i think the formula is f(x)=6x