The joint density function of X and Y is given by f(x, y)=x e^-x(y+1) x>0, y>0 (a) Find the co

step 1 This is the joint density function of x sine y. What we need to find first is the marginal f of

The joint density function of X and Y is given by f(x, y)=x...

Key Concepts

Conditional Density Function

The conditional density function represents the probability distribution of one variable given that a particular value of another variable is observed. It is calculated by dividing the joint density by the marginal density of the given variable. This concept is essential for understanding dependencies between variables and for performing inference based on observed data.

Transformation Technique for Functions of Random Variables

The transformation technique is used to find the probability distribution of a function of random variables, such as the product or sum of them. This involves change of variables and can include methods like using the Jacobian or integrating out other variables. It allows one to derive the density function of new random variables defined by relationships involving the original ones.

Joint Density Function

The joint density function describes the probability distribution of two continuous random variables simultaneously. It assigns probabilities to pairs of outcomes and is the foundation from which marginal and conditional densities are derived. Understanding the joint density is crucial to exploring how the variables interact and how probabilities are allocated over regions in the two-dimensional space.

Marginal Density Function

Marginal density functions are obtained by integrating the joint density function over one of the variables. This process 'marginalizes' the joint distribution, yielding the distribution of a single variable irrespective of the other. It is a key step in further analyses such as finding conditional densities or checking for independence between the variables.

Transcript

00:01 This is the joint density function of x sine y.

00:05 What we need to find first is the marginal f of x.

00:10 The marginal f of x is given by y going from zero to infinity the joint function that means y is zero to infinity x e power minus x into e power minus x y, d .y.

00:27 That is nothing but x, e power minus x into this, we integrate because these two can be taken out because they are not functional y.

00:35 What is integration of e power minus xy? it is e power minus xy divided by negative x and supply the limits is zero to infinity for y.

00:44 So it will be minus e power minus x into when infinity it is zero minus one.

00:51 So it is basically e power negative basically x is an exponential random so the pdf of the random variable x is given by e power minus x when x positive and zero else likewise let us do for y the marginal y is x going from 0 to infinity x e power minus x bracket y plus 1 d x now we need to integrate with respect to x by parts i'll get x into integration of e power minus x into y plus one divided by minus of y plus one minus differentiation of x is one and again we need to integrate this part that part integration is e power minus x into y plus one divided by y plus one but whole is square substitute the limits x as zero to infinity for infinity anyway zero when x is zero this is also zero finally i'll be getting 1 by y plus 1 whole square y positive so this is the probability density function of random be to be y now we need we're interested in the marginals f x y of x x given y is by definition is the joint divided by the margin so that means it is x e power minus x into y plus 1 this is the marginal, sorry, conditional, conditional distribution of x given y.

02:34 And the support of x is x positive.

02:39 So this is the conditional probability density function.

02:44 So what is it? it is x e power minus x times of y plus 1 into y plus 1 whole square when x positive and 0 else.

02:55 How about f y given x of y given x it is f of x comma y divided by f of x so that is nothing but x e power minus x of y plus 1 divided by e power minus x so that will be x e power minus x y now this time the support of the vanamidable y is y positive.

03:18 So that means the conditional period f y given x of y given x is basically x e power minus x y positive and zero else so this is the condition the next thing is we need to find the probability density function of the product of x and y so let us do that first let us start with a cdm capital f z of z by definition it is probability that z is less than it equal to small z that is probability that xy is less than it equal to small z that is c.

03:57 Case 1, if z is negative, then probability that xy less than it equal to z is 0.

04:04 Because x -in -way are positive random variables.

04:06 How can the product be negative? case 2, if z is positive, then we have probability that x, y, less than it equal to z, then we have the region.

04:18 This is the region...

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