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- 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.)6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.
- If X and Y are independent exponential random variables, each having parameter λ.(a) Find the joint density function of U = X + Y by using the convolution of fX and fY .(b) Find the joint density function of V = X − Y by using the method of transformation.(c) Are U and V independent?If X and Y are random variables and X is a geometric random variable where p = 0.1 then what is the probability mass function of Y = sin(X*pi) ?Two random variables X and Y with a joint PDF f(x,y) = Bx+y for 0≤ x ≤6 , 0≤ x ≤7 and constant B Find value of constant B
- 1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.4. A continuous random variable x that can assume value between x = 2 and x = 5 has a density function given by f(x) = 2/27 (1 + x). Find a) P (x < 4) and b) P (3 < x < 4)2.5.8 The random variable X measures the concentration ofethanol in a chemical solution, and the random variable Ymeasures the acidity of the solution. They have a jointprobability density function f(x, y) = A(20 - x - 2y)
- The PDF of a continuous random variable X is as follows: f(X)= c(4x2 - 2x2) 0<* x <* 2 (*less or equal to) a. For this to be a proper density function, what must be the value of c ?For random variables X and Y with joint density function f(x,y) = 6e^-2x-3y. (x,y > 0) and f(x,y) = 0 otherwise, find: Are X and Y independent? Give a reason for your answer.X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
