Joint density functions (gaussian)

In summary, the joint probability density function of two independent standard Gaussian random variables is the product of their individual Gaussian distributions. This can be calculated by multiplying their individual densities together.
  • #1
simba_
19
0
Would just like a hand with this question

If X and Y are independent standard Gaussian random variables (that is, independent N(0, 1) 's ) do the following:
(a) Write down the joint probability density function fXX,Y (x, y) of X and Y .

I know what the gaussian density function looks like. Is it just a matter of multiplying two gaussian distributions together... where u have a σ1 and σ2 (do the same with the mean)
 
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  • #2
Any time X, Y are independent continuous random variables their joint density is the product of their individual densities - so yes, all you need to do is multiply the individual densities.
 

1. What is a joint density function?

A joint density function, also known as a bivariate density function, is a mathematical function that describes the relationship between two random variables in a probability distribution. It shows the likelihood of two variables occurring simultaneously and is typically represented as a two-dimensional graph.

2. What is the difference between a joint density function and a marginal density function?

A joint density function describes the relationship between two random variables, while a marginal density function describes the probability distribution of a single variable. The marginal density function is obtained by integrating the joint density function over all possible values of the other variable.

3. How is a joint density function related to a covariance matrix?

The covariance matrix is a mathematical representation of the joint density function. It contains information about the variance and covariance of the two variables in the joint distribution. The diagonal elements of the covariance matrix represent the variances of each variable, while the off-diagonal elements represent the covariance between the two variables.

4. Can a joint density function be used to calculate probabilities?

Yes, a joint density function can be used to calculate probabilities. The probability of a specific event occurring can be found by integrating the joint density function over the region corresponding to that event. This is similar to how a single variable density function is used to calculate probabilities in a univariate distribution.

5. What is a multivariate normal distribution?

A multivariate normal distribution is a probability distribution that describes the relationship between multiple variables by using a joint density function. It is often used in statistical analyses and modeling to represent data that follows a normal distribution in multiple dimensions. The joint density function for a multivariate normal distribution is typically a Gaussian function.

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