a) The function is f(x), a xb a probability density function. State two properties of this...

 

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a) The function is f(x), a ≤x≤b a probability density function. State two properties of this function [3 marks] and give an expression of obtaining P(a, ≤x≤a) where a ≤a, ≤a, <b b) If X is a continuous random variable with probability density function given by i.) Evaluate k. ii.) Find F(x) and use it to evaluate P(0.3<x<0.6) Jk√x, 0<x<1 f(x)=0, elsewhere c) In a lengthy manuscript, it was found that only 15.5% of the pages has typing errors. Assuming that the number of errors per page follows a Poisson distribution, find the percentage of pages that have exactly one error. [3 Marks] d) The income of 10,000 persons was found to be normally distributed with mean KES 750 per month and standard deviation of KES 50. Show that of this group, about 95% had income exceeding KES 668 and only 5% had income exceeding KES 832. What was the lowest income among the richest 100? [6 Marks] i.) e) A carton containing 50 light bulbs from a production process is checked for faultiness. The mean number of defective bulbs in the box is 5. Assuming that the number of defective bulbs, denoted as X, is a binomial random variable, find P(X ≤2) [4 Marks] f) A continuous random variable Y has a probability density function given by x≤0 ii.) [2 marks] [4 Marks] f(x)= x>0 Show that the moment generating function of Y is given by M,(1)=(1-1)-¹ ||<1 [3 Marks] Use the moment generating function obtained above to find the mean and variance of Y [5 Marks] a) The function is f(x), a ≤x≤b a probability density function. State two properties of this function [3 marks] and give an expression of obtaining P(a, ≤x≤a) where a ≤a, ≤a, <b b) If X is a continuous random variable with probability density function given by i.) Evaluate k. ii.) Find F(x) and use it to evaluate P(0.3<x<0.6) Jk√x, 0<x<1 f(x)=0, elsewhere c) In a lengthy manuscript, it was found that only 15.5% of the pages has typing errors. Assuming that the number of errors per page follows a Poisson distribution, find the percentage of pages that have exactly one error. [3 Marks] d) The income of 10,000 persons was found to be normally distributed with mean KES 750 per month and standard deviation of KES 50. Show that of this group, about 95% had income exceeding KES 668 and only 5% had income exceeding KES 832. What was the lowest income among the richest 100? [6 Marks] i.) e) A carton containing 50 light bulbs from a production process is checked for faultiness. The mean number of defective bulbs in the box is 5. Assuming that the number of defective bulbs, denoted as X, is a binomial random variable, find P(X ≤2) [4 Marks] f) A continuous random variable Y has a probability density function given by x≤0 ii.) [2 marks] [4 Marks] f(x)= x>0 Show that the moment generating function of Y is given by M,(1)=(1-1)-¹ ||<1 [3 Marks] Use the moment generating function obtained above to find the mean and variance of Y [5 Marks]

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1NonNegativity A probability density function is always nonnegative m... View the full answer