QUESTION 1 (a) A random variable X has the following probability density function. f(x) = ke- x , x > 0, k is an unknown constant. (i) Express the constant k in terms of the parameter . (ii) Determine the standard deviation of X using its moment generating function. (b) The number of boys, X, in a family of eight children has a binomial distribution where the chances of a boy are equal to the chances of a girl being a member of the family. Determine the probability of the family having 2 boys or at least 4 girls. (c) Records of a particular open-air kiosk indicate that over a certain period, profits, X, are normally distributed. Further information reveals that the chances of realizing a profit of at least 18.24 are 67% while the chances of realizing at most 27.84 are 97.5% . (i) Determine the mean and standard deviation of the profits generated. (ii) Determine P{|X - | 12} ,where is the mean profit. (d) Determine the number of times a fair die must be tossed so that the probability of the ratio of the number of fives realized to the number of tosses being between 1 /18 and 5 /18 is at least 15/16 .