A point (X,Y) is uniformly drawn from the semi-circle region D: x² + y² ≤ 1, y ≥0 (a) Determine the probability P(Y < X). (b) Determine the Probability Density Function (PDF) f(x) for the marginal distribution of X. (c) Determine the conditional distribution P(X|Y) of X given Y.
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- Let X - U[0, 1]. Find the distribution of Y = 1 - X.Let X be a random variable with uniform distribution on the interval [-2,2]. Let Y be defined as Y = X5. Calculate the pdf of Y.Let X and Y be independent v.a with uniform distribution on the interval (0, 20) and (0, 30) respectively. Find the function f_X+Y (z).
- Suppose that X and Y have a continuous joint distribution for which the joint probability den- sity function is defined as follows: fx,y (x, y) = { vº $4² for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1 otherwise. (a) Determine P(X 1/2). (b) Determine the marginal probability density functions of X and Y. (c) Are X and Y independent? Explain.Let X be a continuous random variable with density function(x) = 4 1. Determine F(5) 2. Determine p(1let fx(x,y)= 1/x, 0<y<x<1 derive distribution of y and compute e(y) and var(y)Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.24. Consider a distribution with density on the interval [0, 2]. Let the probability density function (pdf) for this distribution be the following: fy(u) = % on [0, 2] 2 (i) Draw/plot the pdf fy(y) vs. y for the interval [0, 2]. (ii) Determine the Cumulative Distribution Function (CDF), F, (y). (iii) Draw/plot the CDF F,(y) vs y for the interval [0, 2]. (iv) Determine the expected value of the Random Variable (RV) y, i.e., E [y].