Let X and Y be random variables with μ1=1, μ2=4, σ1^2=4, σ2^2= 6, ρ=(1)/(2). Find the mean and v

Let X and Y be random variables with μ1=1, μ2=4, σ1^2=4,...

Key Concepts

Random Variables

Random variables are functions that assign numerical values to outcomes of a random process. They are characterized by probability distributions, which describe the likelihood of different outcomes, and are fundamental in probability theory and statistics for modeling real-world uncertainties.

Linear Transformation of Random Variables

A linear transformation involves creating a new random variable by multiplying an existing variable by a constant and then adding another constant. This concept is important because it allows for the scaling and shifting of distributions, and it directly impacts the mean and variance of the transformed variable.

Mean of a Linear Combination

The mean of a linear combination of random variables is calculated as the same linear combination of their means. This follows from the linearity of expectation, which states that the expected value operator distributes over addition and scalar multiplication.

Variance of a Linear Combination

The variance of a linear combination of random variables is not simply the linear combination of the variances because variability interacts through covariances. For independent random variables, the variance of the combination is the sum of the variances multiplied by the square of their coefficients; when the variables are not independent, additional terms involving the covariances need to be included.

Covariance and Correlation

Covariance measures the degree to which two random variables change together, while correlation standardizes this measure to a dimensionless quantity ranging between -1 and 1. When computing the variance of a linear combination of dependent random variables, the covariance terms, which can be expressed in terms of correlation and the standard deviations, play a crucial role in accounting for the inter-dependence.

Transcript

00:02 Hi, here it is given that x and y be random variables with mu one equals to 1, mu2 equals to 4, sigma 1 square equals to 4 and sigma 2 squared equals to 6, and row is given equals to half.

00:15 We have to find the mean and variance of the random variable z equals to trice x minus twice 1 okay so two random variables x and y there it is given where expectation x is equals to mu 1 is equals to 1 and variance of x is equals to sigma 1 square is equals to 4 and expectation of y this is equals to mu 2 is equal to 4 and variance of y is is equal to sigma 2 square is equal to 6 these are given and also row is equals to half is given a new random variable let us take z this is equals to trisex minus twice y we have to find the mean and variance of these new random variables that means expectation of z and variance of z okay so with the problem it is given row is equals to half that means correlation coefficient between x and y is half this implies covariance between x and y whole divided by root over variance of x into variance of y this quantity is half now covariance of x this is whole divided by root over variance of x is how much variance of x is four and variance of y is six so four into six equals to half now this implies covariance x comma y this is equals to half into root over four into six that means two root six this is equals to root six we have got covariance between x and y is equal to root 6 right just a minute x comma y yeah now to find the mean and variance of the random variable z so expectation z this is equal to expectation of thrice x minus twice y right now this is expectation of thrice x minus expectation of twice y right now expectation of thrice is a constant so it comes out three into expectation x minus two is a constant two comes out two into expectation y so three into expectation of x is mu one minus two into expectation of y is mu two value of mu one and to 4 right so this will be 3 into 1 3 minus 2 into mu 2 that means 2 into 4 8 3 minus 8 is minus 5 so we have got expectation of z is minus 5 now to find the variance of z see variance of z is equals to variance of thrice x minus twice y right now this is equal as to variance of thrice x plus variance of minus twice y minus two into three into two that means 12 covariance xy right so this is variance of thrice x3 is a constant it is and it is inside the variance so it comes out as a square so three square means nine of x variance of x is sigma 1 square now this minus 2 is a constant and it is inside the variance so comes out as minus 2 square that means 4 into variance of y means sigma 2 square minus 12 into covariance x 1 x comma y you have got root 6 so sigma 1 square sigma 1 square is equals to 4 and sigma 2 square is equals to 6.

05:09 So 9 into 4, that means 36 plus 4 into sigma 2 square is 6, that means 24, minus 12 root 6.

05:20 So 36 plus 24, this is equal to 60 minus 12 root 6...

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