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- 2. Let X be a continuous random variable with probability density function (pdf) 3x�� 0* 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.A variable X is distributed at random between the values 0 and 4 and its probability density function is given by: f(x) = kx³ (4- x)². Find the value of k, the mean and standard deviation of the distribution.4. Let 0 be a parameter. Show that the family of functions Oe-0/x fo(x) = I > 0 is a probability density function. What can you say about the mean of a random variable that has fo as a pdf?A continous random variable has density function f(x)=(3x?)/(63) for Osxse, for some 0>0, and zero otherwise. What is the ratio of the mode to the median for this distribution? A. 2(1/3) B. 0/(21/3) O C.e D. 2(-1/3)3. The random variable a represents the annual rainfall depth in El Paso in inches. Be- tween values of r = 0 and r = 15, the probability density function has the equation 1 f(x) = 15 0Figure I shows the piecewise function (I), (II), (III) and (IV) for cumulative distribution function F(x) for continuous random variable. F(x) (6. 1) IV III II (4.0.8333) (0.0.1667) Figure I Construct the probability density function fix). Should one of the piecewise functions (IV) is not constant, explain the changes.10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.1. 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).7. )The cumulative distribution function of a random variable X is defined by the equation if x1 a. Determine the probability density function of the random variable X b. Determine the probability random variable x from ½ to 1 c. Is the probability density function a valid PDF?Let X be a random variable with pdf f(x) = kx*,-16. Let Yi and Y2 be independent exponential random variables, both with mean 0>0. Let U = - and V = Y, +Y,. Using a change of variable technique, %3D Y, +Y; (a) find the probability density function of U. (b) find the probability density function of V.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,