Let X and Y be independent uniform (0,1) random variables. (a) Find the joint density of U=X, V=

Let X and Y be independent uniform (0,1) random variables....

Key Concepts

Marginalization

Marginalization is the process of integrating a joint density function over one or more variables to obtain the marginal density of the remaining variable(s). This is an essential step when the goal is to find the probability density function of a single variable that is derived from a joint distribution, such as determining the distribution of the sum of random variables by integrating out the other variable.

Joint Density Function

A joint density function characterizes the probability distribution of two or more random variables taken together. When the variables are independent, their joint density is the product of their individual densities. This property simplifies the process of transformation since the initial form of the density is straightforward, making it easier to apply the change-of-variables formula during the transformation process.

Jacobian Determinant

The Jacobian determinant is a crucial element in the change-of-variables formula for transforming joint probability densities. It represents the factor by which volume elements change under the transformation. In the context of transforming variables, calculating the absolute value of the inverse Jacobian ensures that the transformed density function correctly accounts for stretching or compression in the coordinate space.

Transformation of Random Variables

This concept involves finding the distribution of new random variables that are functions of other random variables. When a transformation (for example, U = X and V = X+Y) is applied to a joint distribution, the joint density of the new variables can be determined by employing the change-of-variables technique. This approach is widely used in probability theory to understand how functions of random variables behave and to compute their probability densities.

Transcript

00:01 Hi guys in this problem it's given that n follows a geometric distribution with parameter p okay so we know that f of n when it's equal small n this is q power n times p so the conditional distribution function of x f of x given n is equal to small n follows also a gamma distribution with parameter n and lambda okay so f of x given n is equal to small n follows also a gamma distribution with parameter n and lambda given n is equal to n this is lambda power n x power n minus 1 e power negative lambda x over gamma of n for any x more than 0 okay so it's required to find f of n given x is equal to x okay so this is can be this can be obtained by the function of of n intersect x over the function of x.

01:12 Ok, let's find the function of intersection between n and x.

01:19 This is function of x given n is equal to n times f of n.

01:28 So this is lambda power n times x power n minus 1, times a power negative lambda x over gamma of n times q power n p okay now we need to find uh the function of x when it's equal to x this is integration from zero to infinity for f of x given n times f of n okay so this is integration from zero to infinity for lambda power n times x power n minus 1 times e or negative lambda x over gamma of n times q n p okay so using map software this gives us a limit when n goes to infinity for um lambda n times x power n minus 1 times q power n times e power negative lambda x times e power lambda x minus x times x minus 1 no it's not times we don't have brackets here i'm sorry times x minus 1 over len of q plus len of x plus lmada okay times e power negative lambda x times p over x times gamma of n okay so for simplification we can get this as limit when n goes to infinity for lambda n times x power n minus 1 times q power n times x minus 1 over l n of q plus lenn of n of n plus lena of n plus lena of lambda okay, this is times e -pore -negative lambda x p over x times gamma of n...

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