(a) A continous random variable X has a pdf given by k(1 1), 1x 0 f(x) 2k(1 x), 0x1 elsewhere 0, (i) Find the value of the constant k (ii) Obtain the standard deviation of the random variable X X (b) The table below shows the number of days it takes a student to report to university for the new semester after a short break The number of days, X, is a random variable with probability mass function given below 1 2 3 0 18 b a 4 0 22 P(X x) (i) If E(X) 3 05, determine the value of a and b (ii) Find the standard deviation of the random variable X 5 0 2 (1) 2 marks 5 marks (i) Mutually exclusive events (ii) Indepentent events (c) The data below gives the weights of a group of patients under medication for COVID 19 Disease 31 1, 41 3, 52 7, 62 7, 33 2, 47 8, 55, 68, 64, 38, 9, 45, 55 8, 32 1, 47, 61 7, 36 8, 49, 2, 65, 40 1, 60 2, 48, 66 1, 47 5 4 marks 3 marks (i) Construct a grouped frequency distribution table (ii) Using assumed mean method, calculate the mean weight of the patients (d) Family income is believed to be normally distributed with mean of kshs 25,000 and a stan dard deviation of kshs 10,000 if poverty level is 10,000, what percentage of the population lives in poverty A new tax law is expected to benefit the middle income families, those with income between kshs 20,000 and kshs30,000 what percentage of the population will benefit from the law 5 marks 2 (e) The mean mark of 14 students from a certain population is 64 7 If the standard deviation of the population is 3 67 Construct a 95 confidence interval 4 marks 6 marks 3 marks (f) Given that two events A and B are such that P(A) 0 5 and P(AUB) 0 71 find P(B) if A and B are 2 marks 3 marks

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(a) A continous random variable X has a pdf given by k(1+1), 1x <0 f(x)= 2k(1-x), 0x1 elsewhere 0, (i) Find the value of

(a) A continous random variable X has a pdf given by k(1+1), 1x

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a i Find the value of the constant k Since the pdf has to integrate to 1 over its support 0 x 1 we can use this property to find the constant k k integral from 0 to 1 of 21x dx 2 x 12x2 from 0 to 1 2 ... blur-text-image

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