Suppose that the joint density of X, Y is 1A where A is the square with corners at (0,1),(1,0),(

Suppose that the joint density of X, Y is 1A where A is the...

Key Concepts

Joint Density Function

The joint density function describes the probability distribution of two random variables taken together. It defines, for each pair of values, the probability density that the variables simultaneously take those values, and it must integrate to one over the entire space where the random variables are defined.

Support of a Probability Density Function

The support of a probability density function is the region in the sample space where the density is non-zero. In this context, it is the set of all points where the indicator function equals one. Understanding the support is crucial for computing probabilities and evaluating properties such as independence.

Independence of Random Variables

Two random variables are independent if the joint density function can be written as the product of their marginal density functions. This means that the behavior of one variable does not affect the other. Verifying independence typically involves computing the marginals and checking if their product equals the joint density function for all values in the support.

Marginal Density Functions

Marginal density functions are obtained by integrating the joint density function over the range of the other variable. They represent the individual probability distributions of each random variable irrespective of the other. The product of the marginals is used to determine if the joint density factors as required for independence.

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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