Var(∑AiYi): Calculating & Explaining the Formula

[milk]

Var( ∑AiYi)= ∑(Ai^2) Var(Yi)
Could you show why?
Thank you

 

[mathman]

Your equation includes an assumption that the Y's are independent random variables. In this case, you only need to show it for one variable, i.e. var(AX)=A²var(X). This then can be shown by using the definition of var in terms of first and second moments.

 

[tacman]

for your question. The formula for calculating the variance of the sum of random variables, Var(∑AiYi), can be explained using the properties of variance and covariance.

First, let's break down the formula into two parts: ∑(Ai^2) and Var(Yi). The first part, ∑(Ai^2), represents the sum of the squared coefficients for each random variable, Ai. This is important because it takes into account the relative weights or contributions of each random variable to the overall sum. For example, if one random variable has a larger coefficient than another, it will have a greater impact on the overall sum and therefore, its squared coefficient will be larger.

The second part, Var(Yi), represents the variance of each individual random variable. This is the measure of how spread out the values of that random variable are from its mean. So, by multiplying the squared coefficients with the variances of each random variable, we are essentially weighting the variances based on their contributions to the overall sum.

Now, let's look at why this formula works. The variance of a sum of random variables can be calculated using the following formula: Var(∑Xi) = ∑∑Cov(Xi,Xj), where Cov(Xi,Xj) represents the covariance between two random variables.

Using this formula, we can expand Var(∑AiYi) to: Var(∑AiYi) = ∑∑Cov(AiYi,AjYj).

Since the random variables Ai and Aj are independent, their covariance will be equal to 0. This means that the only non-zero terms in the summation will be when i = j, resulting in the formula: Var(∑AiYi) = ∑Var(AiYi).

Now, we can further expand Var(AiYi) to: Var(AiYi) = E[(AiYi)^2] - [E(AiYi)]^2. By substituting this into the previous formula, we get: Var(∑AiYi) = ∑[E((AiYi)^2) - [E(AiYi)]^2].

Finally, we can use the properties of variance to simplify this further to: Var(∑AiYi) = ∑(Ai^2)Var(Yi), which is the original formula we started with.

In summary,