Consider f(x) = e, x>0, and joint probability density function fxy(x,y)=e, for 0 3 Of Y13=e³-³, y<3 Of y13=e³-y, y>3
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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .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.4.6-3.) Random variables X and Y have respective density functions $x(x) = -[u(x) – u(x-a) to fy(y) = bu(y)e-by ob where a > 0 and b>0. Find and sketch the density function of W = X + Y if X and Y are statistically independent
- 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?Pb(Z = z) = B (8.25z - z2) with z E {1, 2, ......., 5}. What value must B be to ensure the probability mass function for Z, equation(1), is valid?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) ?
- QUESTION 11 The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a ≤ x ≤ b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices.Suppose that the random variables X, Y , and Z have the joint probability density function f(x,y,z)=cxyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Find the E(x). Use the Scientific Method of Answering (Given, Required, Formula and Solution.)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?
- Question c) Let X = [X1,X2, Xg] be a three dimensional random vector uniformly distributed on [0,1]x[0,1 ]x[0,1]. Let Y= g(X ), be a two dimensional random vector, where 91X)=X, + Xz. 92(X ) = X2 X3 . Find the density function of 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.)) If ρ : [0, π/2] → R, ρ(x) = c(sin(3x) + 1) is the probability density function of a continuous random variable X, find the constant c. b) Compute M(X) and D(X). c) Compute P(X ≥ ln 2). Plz explain a) and b)