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

understand that i don't know the answers

will someone answer his question

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.