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- (3) Let X = b >(8,-) find E(5+6x) and distribution function.Q2) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the below density function , Find the probability that a bottle of this medicine will have a shelf life of anywhere between 60 and 120 days 20000 0Let the density function of a random variable X be given by fx(x) = 0x®-', 01. Let f be the probability density function of a continuous random variable X whose range is (-0, 00), where if a sxsb f(x) ={32 0 otherwise. (a) Explain why a, hence b, must be nonnegative. Graph the function f. (b) Find the value of b'-a. 7 (c) If it is known that P(X 8).10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.For the continuous probability function f (x) = Kx² e* when x20 fint i) k i) mean iii) varianceSuppose that the random variable X has the probability density function for -1s x < 1 c(1– x?) f(x) = elsewhere Find the value of the constant c.Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).Let X be a random variable with pdf f(x) = kx*,-1* Consider the probability density function fx (x) = a e-b lel where X is the random %3D variable which assumes all the values from (i) relation between a and b -00 to -00, Find (ii) the probability of finding X in the range 1 to 2.Suppose that the random variables X and Y have the following joint probability density function. f(x, y) = ce-8x - 5y, 0 0).7) Allow a continuous probability distribution to be defined as f(x) = x/2 for the range 0 sxs2. Demonstrate that the area underneath the curve is 1 (the function integrates to 1) and calculate the mean and variance of the probability distribution.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,