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Q: [1] The probability density function of a random variable X is = {c(x+4), -4 4.
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A: Hey there! Thank you for posting the question. However, since your question has more than 3 sub…
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Q: [1] The probability density function of a random variable X is SC(x + 4), -4 4.
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- 5. Suppose that the joint probability density function of X and Y is given by f(x, y) = C (x² + y), 0 < x < 2, 0 < y < 2 and f(x, y) = 0 otherwise. (a) Find C. (b) Find the marginal density functions of X and Y. (c) Compute P(X > Y). (d) Compute E[X]. (e) Compute E[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.)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.
- Suppose X, Y, and Z are random variables with the joint density function f(x, y, z) = Ce−(0.5x + 0.2y + 0.1z) if x ≥ 0, y ≥ 0, z ≥ 0, and f(x, y, z) = 0 otherwise. (a) Find the value of the constant C.(b) Find P(X ≤ 0.875 , Y ≤ 1.125). (Round answer to five decimal places).(c) Find P(X ≤ 0.875 , Y ≤ 1.125 , Z ≤ 1). (Round answer to six decimal places).1)Let x and y be two continuous random variables whose function is the probability density joint is given by : a)Draw the relationship between the variables x and y on the Cartesian axes .b)Calculate the marginal pdfs px(X)and py(Y).c)Are the v.a.s x and y independent ?Let Y1 and Y2 be random variables with joint density functionf(y1, y2) = (6/7(y1^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).c) Find P(0.6 < Y1 < 1|0 < Y2 < 1).
- 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 independentSuppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?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) = ______________________
- 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).Let f(x, y) = C/8, 0 ≤ y ≤ 4, y ≤ x ≤ y + 2, be the joint pdf of X and Ya) Find the value of C so that f(x,y) is a valid joint p.d.fb) Find the marginal probability density function of X, fX(x)c) Compute μX , μY , Cov(X, Y), and ρ.Suppose that two continuous random variables X and Y have joint probability density function fxy = A( ex+y + e2x+y) , 1 ≤ x ≤ 2 ,0≤ y≤3 0 elsewhere a. P ( 3/2 ≤ X ≤ 2, 1 ≤ Y ≤ 2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0
