91Ó°ÊÓ

Variance is a measure of how much a random variable, such as \(X\), spreads out from its expected value. It gives us an idea of how values in a data set are distributed around the mean. Calculating variance involves understanding both the expected value \(E(X)\) and the expected value of the square of \(X\), which is \(E(X^2)\).
To calculate variance, the formula we use is:

• \(V(X) = E(X^2) - (E(X))^2\)

The variance formula tells us that we start with the expected value of the squared outcomes, \(E(X^2)\), and subtract the square of the expected value, \((E(X))^2\).
Here's a simple way to see it: You’re looking at the average of squared distances from the mean. In our example:

• \(E(X) = 5\)
• \(E(X^2) = 32.5\)

Substituting these into the variance formula, we get:

• \(V(X) = 32.5 - 5^2 = 32.5 - 25 = 7.5\)

This means the variance, \(V(X)\), is 7.5, which shows the average squared deviation of the data from the mean.

The relationship formula links different statistical measures of random variables. It helps to understand how various expectations interact and can be manipulated to derive unknown quantities.
One such useful relationship presented in this exercise is:

• \(E[X(X-1)] = E(X^2) - E(X)\)

This equation stems from an identity: multiplying \(X\) by \(X-1\) gives \(X^2 - X\). Thus, the expectation, \(E\), applies similarly like arithmetic operations:

• \(E[X(X-1)] = E(X^2 - X) = E(X^2) - E(X)\)

Using this, we could solve for \(E(X^2)\) if given \(E(X)\) and \(E[X(X-1)]\). Plugging our values in:

• \(27.5 = E(X^2) - 5\)

Adding 5 to both sides to find \(E(X^2)\):

• \(E(X^2) = 32.5\)

This demonstrates how the formula allows determination of a missing expectation from known values.

Random variables are foundational to probability and statistics, representing numerically-valued outcomes of random phenomena. They can be thought of as variables whose possible values are numerical outcomes of a random phenomenon.
We encounter different types of random variables:

• Discrete Random Variables: These have countable outcomes, like rolling a die or flip of a coin.
• Continuous Random Variables: These have an infinite number of outcomes, such as measuring time or weight.

In working with random variables, concepts like the expectation and variance are crucial. The expectation, \(E(X)\), also known as the mean, tells us the average outcome if an experiment is repeated many times. Variance, as discussed, measures how spread out the values of the random variable are from the expectation.
The understanding of random variables and these measures forms the backbone for statistical inference, allowing us to make predictions and analyze patterns from data.