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- The random variables X and Y have the following joint probability density function:f(x,y)={e−x−y , 0<x<∞; 0, elsewhere. What is Cov(X,Y)(X,Y)?Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentThe 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 ?
- Calculate the E(X) when the joint probability density function of X and Y is fxy(X,Y)=c(X+Y) over the range x = 1, ..., 4 and y = 1, ..., 2The joint probability density function of X and Y is given byf(x,y) = x+y 0 < x < 1, 0 < y < 1 (d) E(xy) = ? (e) Cov(x,y) = ?Let X be a random variable with density function f(x) = cx−3, if x ≥ 1, 0 otherwise. a) Find c.b) Find P (3 < X ≤ 6).c) What is P(X = 3)?
- The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function, f(x) = 0.05x + 0.3 if 3 <= x <= 5, and 0 otherwise. (a) Calculate P(X ≤ 4). (b) Calculate P(3.5 ≤ X ≤ 4.5). (c) Calculate P(4.5 < X).The random variables X and Y have a joint density function f(x, y). f(x,y)= 2x+2y 0<x<1, 0<y<1, y>x 0 elsewhere P(X < 0.5, Y < 0.5) = ______________________The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function. What is P(X ≤ 4)? What is P(4.5 < X)?
- The proportion of people who respond to a certain mail-order solicitation is a random variable X having the following density function. f(x) = 2(x+1)/3, 0<x<1, 0, elsewhere Find σ2g(X) for the function g(X)=5X2+4. σ2g(X)= ?b) Given the following density function of the random variable X: f(x) = (1+x)2, 0<x<1 = 0, Find E(X) and V(X).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.)
