Two random variables X, Y are known to be jointly Gaussian, so their joint densi

Question

Two random variables X, Y are known to be jointly Gaussian, so their joint density function has the f_XY = 1/2 pi Squareroot 1 - rho_XY^2 exp(-1/2(1 - rho_XY^2 exp(-1/2(1 - rho_XY^2) [(x - mu_X)^2/sigma_X^2 + (y - mu_Y/sigma_Y^2 - 2 rho_XY (x- mu_X) (y - mu_Y)/sigma_X sigma_Y]) where mu_X and mu_Y are the means of X and Y respectively, sigma_X and sigma_Y try are the standard deviations (the squareroot of the variance) of X and Y respectively, rho_XY is the correlation coefficient between X and Y, and Lambda_XY is the covariance matrix of X and Y. Suppose mu_X = 2, mu_Y = 3, Lambda_XY = [1 -1 -1 9], and we define Z = 3X - Y + 2. Find the density function for the random variable Z.

Explanation / Answer

The mean of a random variable is defined as the weighted average of all possible values the random variable can take. Probability of each outcome is used to weight each value when calculating the mean. Mean is also called expectation (E[X])

For continuos random variable X and probability density function fX(x)

E[X]=xfX(x)dxE[X]=xfX(x)dx

For discrete random variable X, the mean is calculated as weighted average of all possible values (xi) weighted with individual probability (pi)

E[X]=X=xipiE[X]=X=xipi

Variance :

Variance measures the spread of a distribution. For a continuous random variable X, the variance is defined as

var[X]=(x–E[X])2fX(x)dxvar[X]=(x–E[X])2fX(x)dx

For discrete case, the variance is defined as

var[X]=2X=(xi–X)2pivar[X]=2X=(xi–X)2pi

Standard Deviation () is defined as the square root of variance 2X2X

Properties of Mean and Variance:

For a constant – “c” following properties will hold true for mean

E[cX]=cE[X]E[cX]=cE[X]

E[X+c]=E[X]+cE[X+c]=E[X]+c

E[c]=cE[c]=c

For a constant – “c” following properties will hold true for variance

var[cX]=c2var[X]var[cX]=c2var[X]

var[X+c]=var[X]var[X+c]=var[X]

var[c]=0var[c]=0

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