The joint probability density function of a pair continuous...

The joint probability density function of a pair continuous random variables X and Y is given by f(x, y) =    1 8 (6 − x − y) for 0 < x < 2, 2 < y < 4 0 elsewhere (a) Find the marginal densities of X and Y , respectively. (6 points) (b) Find the conditional density of X given Y = y, and use it to evaluate P(X ≤ 1|Y = 3). (8 points)

Transcript

00:02 Hello, fxy we are given the joint probability distribution of random variables xy as fxy is equal to 12 times e to the power minus 4x minus 3y when x is non -negative and y is non -negative and it is equal to zero otherwise and the first thing we please note that fxy we can write as 12 times the product of e to the power minus 4x and e to the power minus 3y when x and y are both non -negative and equal to zero otherwise we can split this thing as a product of these two now we calculate the marginal probability distribution of x so it's whose value at any x is given by the integral of fxy dy from minus infinity to infinity when x is less than zero fxx x is equal to integral of but when x is less than zero fxy is zero so fxx equal to integral of fxy dy which is equal to integral of zero dy hence it is equal to zero so when x is less than zero f the marginal probability distribution of x is equal to zero when x is greater than equal to zero the marginal probability distribution of x is the integral from minus infinity to infinity fxy dy but fxy is equal to zero when y is negative so this integral is equal to the integral from zero to infinity fxy dy now in this region fxy is equal to this thing 12 into e to the power minus 4x e to the power minus 3y dy integral from zero to infinity now note that the integral of well we can take 12 into e to the power minus 4x out because they are like constant for this integration and we are we have to integrate zero to infinity e to the power minus 3y dy now e to the power minus 3y dy from zero to infinity is equal to one over three so we so what we get is 12 into e to the power minus 4x times one over three three cancels and we get four so it is four times e to the power minus 4x so we get that f the marginal probability distribution of x is zero when x is less than zero and it is four into e to the power minus 4x when x is greater than equal to zero similarly let us go on to let us find the marginal probability distribution of y it is equal to integral of fxy with respect to dx from minus infinity to infinity when y is less than zero fxy is zero so the marginal probability distribution of y is integral of zero dx from minus infinity to infinity which is zero when y is greater than equal to zero this fyy is equal to integral from minus infinity minus infinity to infinity fxy dx but this is zero when f y x is negative so this integral is equal to integral from zero to infinity fxy dx now fxy in this region is equal to this thing 12 e to the power minus 4x e to the…

Question

Please give Ace some feedback