a) A continuous random variable has the following probability density function. f(x) = (1 /12)(2x + 1) , 0 < x c, = 0, otherwise. Determine the value of the constant c and hence compute E(X2 ). (2+2) mks

b) A certain company makes solar bulbs. Records indicate that, on average, these bulbs remain bright for ten consecutive hours with a probability of 0.75. Six bulbs are picked at random.

i) Develop a p.d.f. for the bulbs that would remain bright for ten hours. (3 mks)

ii) Compute the probability of four or none of them remaining bright for the stated period.(3mks)

c) The mean and variance of a continuous variable X are 35 and 25 respectively. Determine:

(i) The value of the constant c such that P{|x - 35| c} 1 /16 . (3mks)

(ii) The minimum value of P(10 x 60). (3mks)

d) Two discrete random variables have the following joint probability density function. f(x,y) = k(y2 x + 2yx + 1) , x = 0, 1 ; y = 1, 2 = 0, otherwise. Determine the value of the constant k and hence compute f(x 2, y 1). (2 + 2)mks

e) Consider the following probability density function of the random variable X. f(x) = 0.5e- 0.5x , x > 0 (5mks)

Using cumulative distribution function technique, determine the density function of Y = 4X.

f) Monthly revenues (Y) and corresponding expenditures (X) for a large-scale business are positively correlated. The regression line of expenditure on revenue passes through the point (x=0, y=0). For the last twenty months, mean expenditure and mean revenue were ksh206.4M and ksh240M respectively.

Determine the best estimate of the expenditure corresponding to revenue of ksh300M. (5mks)