Q1. a) Consider the following probability density function of the random variable X.
X 6 10 8 k
f(x) 2/10 4/10 3/10 1/10


(i) Given that the mean of X is 8, determine the value of the constant k.
(ii) Compute the standard deviation of the random variable X.
b) Potassium Phosphate is one of the components of DAP fertilizer. The amount of Potassium
phosphate, X (in kg) for a 100kg bag varies with the type of soil where this fertilizer has to be used.
Research reveals that the probability density function of X is as follows:
f(x) = (1/14)(3x2 + 2x + k) , 0 ≤ x ≤ 2 ,
0, otherwise
(i) Determine the value of the constant k. (ii) A bag was picked at random. Determine the probability
of the amount of potassium phosphate being at least 1.0 kg