a) i) it is a poisson distribution. This is so because it defines the probability of a number of independent events occurring in a fixed time. It shows how many times an event is likely to occur over a specified period. ii) x is a poisson variable with mean = 3, so the probability mass function of of x is p(X=x) = (e-3 * 3x)/(x!), x= 0,1, 2,3, ...... iii) The possible values of x are 0,1,2,3,4 .... iv) we define p(x≥3) = 1- p(x=0) - P(x=1) - P(x=2) = 1- ( (e-3 *30) - (e-3 * 31) - ( (e-3 * 32) /2) ) =0.5767 which is the required solution.

a) we define X as the number of houses in urban areas that were burglarized.

There are 50 houses taken as a sample and the probability that has a house will be burglarized is 0.05, which we may define as the probability of success.

Thus x ~ Bin(50,0.05)

b) we define the pmf of x as p(X=x) = $\begin{pmatrix} 50 & \\ x& \end{pmatrix}$ (0.05)x(0.95)50-x x= 0,1,2 . . .. ,50

$0, otherwise$

so x can take values 0, 1, 2, 3, 4, 5, . . . . . ,50

c) E(x) = (50 * 0.05) = 2.5 which is approximately three houses.

d) standard deviation = (npq)1/2

d\given p = 0.05, q= 1-p = 1-0.05 = 0.95

Thus standard deviation = (50 * 0.05 * 0.95)1/2

= 1.541103501

e) we define p(x=0) = ( (50C0) * (0.05)0 * (0.95)50 )

= 0.076944975 which is the required solution.