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- 2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)Find the mean and variance for the distribution of random variable X whose density function is f(x). f(x,y)= 1/16x^2e^-x/2 x>0Find the conditional expectation E(Y/X=0.47) if the joint probability density function of the random variable X and Y isf(x, y) = 1/x, 0 < y ≤ x ≤ 1.
- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.A continuous variable Y has a probability density function for which the moment generating function is given by M(t)=e^(2*t+72*t^2). What is the variance of the variable, Var[Y]?For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
- A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 01)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)variance
- Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.A random variable X has probability density function (pdf) fx(x), wherefx(x)= c(1–x4) -1≤x≤1elsewherei)Find the cdf Fx(x)of Xii)Find ciii)Find the variance of X.Since P{X=1}=P{X=2}=P{X=3}=0.2 , P {X=4}=0.4 , plot the probability density function of the random variable X and the distribution function.