Determine the value for c and the covariance and correlation for the joint probability density f

Determine the value for c and the covariance and correlation...

Key Concepts

Correlation

Correlation standardizes the measure of the linear relationship between two random variables by dividing the covariance by the product of their standard deviations. This results in a dimensionless value ranging from -1 to 1, which allows for a comparison of the strength and direction of the relationship regardless of the scale of the variables.

Covariance

Covariance is a measure that quantifies how two random variables vary together. It is defined as the expected value of the product of their deviations from their respective means. A positive covariance indicates that the variables tend to increase together, while a negative value indicates an inverse relationship.

Joint Probability Density Function

A joint probability density function (pdf) assigns probabilities to combinations of outcomes for two or more continuous random variables. It must be non-negative over its entire domain and its integral over that domain must equal one, ensuring that the total probability is correctly normalized.

Normalization Constant

The normalization constant is a scalar multiplier used in a probability density function to guarantee that the integral of the function over its entire support is equal to one. Finding this constant typically involves integrating the function over its specified range and solving for the constant that makes the total probability one.

Integration over Specified Domains

Integration is a key technique used in probability to calculate the probabilities, expected values, and moments of continuous random variables. When the region of integration is defined by constraints (such as a variable dependent region), setting correct limits for the integration is essential to correctly capturing the entire support of the distribution.

Transcript

00:01 Hi guys, in this problem we are given that is a showing density function of x and y such that f of x and y equals c x y where x where x is between 0 and 3 and y between 0 and x and otherwise okay so we have to find the value of of c such that the given density is a proper density and we know that the integration over x and y for f of x and y equals 1 so integration for x from 0 to 3 then integration of y from 0 to x for c x y okay d y x equals 1 so this integration integration from 0 to 3 for cx then y squared over 2 then we substitute by its limits x and 0 d x so this equals to c over 2 integration from 0 to 3 for execute the x okay, so this equals to c over 2 times x power 4 over 4 and then substitute by elements of x, which is 3 and 0.

01:45 So this equals to c over 2 times 3 power 4 over 4 equals 1.

01:54 And hence c equals 8 over 81.

02:03 Okay.

02:05 Now the preferred density function is 8 over 81 xy between the given limits.

02:11 So now we have to find the covariance and correlation between x and y.

02:19 So let's, the first step we need to do is to find the mean of x, which is integration.

02:26 From 0 to 3, then integration from 0 to x, 4 ,8x times f of x, which is 8 over 81, xy, d -y, d -y, d -x.

02:45 Okay, so calculating this integration, we get 8, 4, sorry, it's 4 over 81 times the integration from 0 to 3, 4, 4 ,000, times the integration from 0 to 3, 4x.

02:59 Power 4 the x.

03:02 Okay, so this is 4 over 81 times x power 5 over 5...

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