Q1- A survey show that 60%% of the housewives have seen the advertisement of a new product in televisions The probability that a housewife who has seen the advertisement buys the new product is 0.9 while the probability that a housewife has not seen the advertisement buys the new product is 0.3

(a) Find the probability that a housewife buys the new product?

(b) Fin d the probability that a housewife who buys the new product h as seen the advertisement?

(c) Using appropriate approximation find the probability of less than 10 of 50 housewives who buy the new product have not seen the advertisement?

Q2- a) The height of students at a local university are normally distributed with a mean of 168 cm and standard deviation of 8 cm. One of these students is chosen at random. Find the probability that the height of the student is :

i.. Less than 172 cm.

ii. Between 160 cm and 170 cm.

b) based on (a) given that 30% of all these students have height less than H cm, find the value of H.

(c) Based on (a) If the an adjustment can be made to the number of students which alters the mean but leaves the standard deviation unchanged, what should the new mean height be if it is required that 95%% of the height less than 170 cm .

Q3- find the following:

(a) Find the mean and variance from the given moment generating function

M_x(t) = e to the power (e^t-1) and M_x(t) =e to the power of t+1/2 ² t²

(b) If the exponent of e of a bivariate normal density is

-1/54 [x²+4y²+2xy+2x+8y+4]

Find _{1 2} and , given that mu 1=0 mu2 =-1

(c) A panel of prospective jurors includes six married men, three single men, seven married women, and four single women. If the selection is random, what is the probability that a jury will consist of four married men, one single man, five married wo men and two single women?

Q4- Give the function f(x,y) = 4xy, 0

(a) Determine the marginal probability of random variables Y.

(b) Determine the conditional probability distribution of X given Y=1/2

(c) Determine E(X/Y=1/2)

(d) Determine²_x/y=1/2

Q5- A discrete random variable X can only takes the values 0,1, 2 and 3 with probability function given by: f(x)= k/2+x , x=0,1,2,3

a) The value of k

b) P(X=2 or X=3)

c) F(X)

d) The mean and variance of X

e ) E(-3X+2) and V(-3X+2)