35. Covariance of Random Variables

Def1. A random variable is a function what associated real number with each element in a sample space.

Def2. The set of ordered pairs (X, f(x)) is a probability function or probability math function or probability distribution of a discrete random variable X for each possible outcomes:

1. f(x) ≥ 0

Def3. Let X is a random variable with probability distribution f(x). The mean or expected value of X is µ = E(X) =

Def4. Let X is a random variable with probability distribution f(x) and expected value µ. The variance of X is equal If the variance is positive root, we called the standard deviation of X. If the negative we study the variance of probability distribution of X, there we denote from

Def5. Let X be a random variable with probability function f(x) and mean or expected value µ. The variance of X is equal

Def6. Let X and Y random variables with joint probability distribution f(x,y). the covariance of random variables X and Y we denote that . There X and Y are discrete.

Def7. Let X and Y random variables with covariance and standard deviation and respectively a correlation ρxy is equal ρxy= .

Remark1. There x and y are free version of covariance called is correlation coefficient and used widely in statistic.

36. The correlation coefficient.

Def1. A random variable is a function what associated real number with each element in a sample space.

Def2. The set of ordered pairs (X, f(x)) is a probability function or probability math function or probability distribution of a discrete random variable X for each possible outcomes:

1. f(x) ≥ 0

Def3. Let X is a random variable with probability distribution f(x). The mean or expected value of X is µ = E(X) =

Def4. Let X is a random variable with probability distribution f(x) and expected value µ. The variance of X is equal If the variance is positive root, we called the standard deviation of X. If the negative we study the variance of probability distribution of X, there we denote from

Def5. Let X be a random variable with probability function f(x) and mean or expected value µ. The variance of X is equal

Def6. Let X and Y random variables with joint probability distribution f(x,y). the covariance of random variables X and Y we denote that . There X and Y are discrete.

Def7. Let X and Y random variables with covariance and standard deviation and respectively a correlation ρxy is equal ρxy= .

Remark1. There x and y are free version of covariance called is correlation coefficient and used widely in statistic.

37. Means and Variances of Linear Combinations of Random Variables. We are still working towards finding the theoretical mean and variance of the sample mean:

Xˉ=X1+X2+⋯+Xnn

If we re-write the formula for the sample mean just a bit:

Xˉ=1nX1+1nX2+⋯+1nXn

we can see more clearly that the sample mean is a linear combination of the random variables X₁, X₂, ..., X_n. That's why the title and subject of this page! That is, here on this page, we'll add a few a more tools to our toolbox, namely determining the mean and variance of a linear combination of random variables X₁, X₂, ..., X_n. Before presenting and proving the major theorem on this page, let's revisit again, by way of example, why we would expect the sample mean and sample variance to have a theoretical mean and variance.