Question

A bad marksman takes 10 shots at a target with probability of hitting the target with each shot of 0.1, independent of other shots. Z is the random variable representing the number of hits.

a) Calculate and plot the PMF of Z.

b) Calculate and plot CDF of Z. (You may desire to manual adjust the plot for our convention)

c) What is the probability that 3 < ?? < 5 shots were hits

d) Find E[Z] and var[Z]

e) If the marksman were to get $x for each shot on target. How much should the marksman expect to get in order to break even on the $10 entry fee?

Answer 1

P(hit) = 0.1

P(no hit) = 0.9

If Z is the random variable representing the number of hits and also each shot is independent of the other shots played we have:

p(Z=1) = 0.1

P(Z=2) = 0.1*0.1 = 0.0121

the pmf of Z is:

P(Z=z) = 0.1^z and P(Z=0) = 1 - (1 - 0.1^10)/(1-0.1) = 0.888889

b) the cdf of Z is :

Z	P(Z<=z)
0	0.888889
1	0.988889
2	0.998889
3	0.999889
4	0.999989
5	0.999999
6	1
7	1
8	1
9	1
10	1

c) I am assuming that th equestion is asking for the probability that 3<=z<=5

P(3<=z<=5) = P(Z=3) + P(z = 4) + P(Z = 5) = 0.00111

d) E(Z) = sum ( Z*P(Z=z) ) = 0.123457

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

E(Z^2) = 0.150892

V(Z) = 0.13565

e) The expected income from each shot is $x

And the expected number of shots made by the marksman are 0.123457

Therefore, in order to cover the fee of $10 the marksman needs:

10/0.123457 = 80.99986 or $81 approximately