Prove that σx y=I y-𝐓 y̅, where x̅ and y̅ are the means of {xi} and {yi}, respectively, while x

Prove that σx y=I y-𝐓 y̅, where x̅ and y̅ are the means of...

Key Concepts

Mean

The mean, or average, is a fundamental concept in statistics that represents the central tendency of a data set. It is calculated by summing all the individual values and then dividing by the number of values, providing a single value that summarizes the data.

Covariance

Covariance is a statistical measure that describes the extent to which two variables change together. It is calculated by taking the average of the product of the deviations of each variable from their respective means, and it quantifies the linear relationship between the variables.

Linearity of Expectation

The linearity of expectation is a key property in probability and statistics stating that the expected value of the sum of random variables is equal to the sum of their expected values. This concept allows us to break down complex expressions into simpler parts during proofs and derivations.

Algebraic Expansion

Algebraic expansion involves distributing products over sums, which is essential when transforming expressions in proofs. In the context of covariance, expanding the product of differences from the mean leads to terms that simplify to the expression E(xy) - E(x)E(y), thereby proving the desired result.

Transcript

00:02 Hi, here it is given that the independent variables x1 and x2 have means mu 1 and mu 2 and variance is sigma 1 square and sigma 2 respectively.

00:14 And you have to show that that the mean and variance of the product y which is equals to x1 into x2 are mu 1, sigma 1 square, sigma 2, plus mu 1 square plus mu 2 square plus i'll write it down two random variables x1 and x2 x1 just a minute x1 x2 these two random variables are independent okay so clearly as they are independent i write correlation coefficient between x1 and x2 is equals to 0 right so that means covariance x1 x1 x2 is 0 so from here expectation of x2 x1 into x2 this quantity is equals to expectation of x1 into expectation of x2 as they are independent so expectation of product of two independent random variables is equals to product of individual expectations and it is given that the random variable x1 has mean mu so expectation of x1 is equals to mu 1 and variance of x1 is equals to sigma 1 square.

01:36 Similarly, expectation of x2 is equals to mu2 and variance of x2 this quantity is equals to sigma 2 square.

01:49 But we have to find we have a new random variable y which is equals to x1 into x2 and we have to show that the mean of this random variable that is expectation of y is equals to mu 1, mu, 2, and variance of y is equal to sigma 1 square, sigma 2 square plus mu 1 square, sigma 2 square plus mu 2 square, prove then.

02:31 Now we have a new random variable which is y equals to x1 x2.

02:35 Clearly expectation of y is equal to expectation of x1 into x2 clearly as they are independent.

02:46 So i have already shown that expectation of x1 into x2 is equal to expectation of x1 into expectation of x2.

02:54 So expectation of product of these two random variables is equal to product of individual expectations.

03:01 That means expectation of x1 into expectation of x2.

03:07 Now expectation of x1 is mu 1 and expectation of x2 is mu 2 from here and here.

03:14 This is equals to mu 1 into mu 2.

03:18 So we have got expectation of y is equals to mu 1 into mu 2...

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