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- Consider the density function f(x) = k√√x, 0The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.Let f(x)= 4e-** if 0Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function below. f(x) = 2(x+13) 0Let X denote 0.025 × the ambient air temperature (˚C) and let Y denote the time (min) that it takes for a diesel engine to warm up. Assume that (X, Y) has joint probability density function f(x,y) = 1.6x (1 − x)(6 + 5x − 4y), for 0 < x < 1, 0 < y < 0.5. Find the regression curve for X as a function of warm-up time Y = y and plot it. Use it to predict the air temperature = 40X for warm up time Y = 0.25 min.Find the total mass of a 1 meter rod whose linear density 39 function is p(x) kg/m for 0 ≤ x ≤ 1 (x + 1)² Mass = = kgDetermine whether f(x) is a probability density function on the interval [0, 4]. If not, determine the value of the definite integral. f(x) = 3x² 256If the probability density of X is given by f(x)=6x(1–x) 0Consider the density function: f(x) = {k√x, 0, (a) Evaluate k (b) Find F (x) and use it to evaluate P (0.3 < X < 0.6) 0 < x < 1 elsewhereSEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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