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# ACMMM140

## 2.3 Variance and Standard Deviation

#### Variance

• The actual outcome will often differ from the mean (or E(X)). Sometimes the difference is large, sometimes it is not (or even sometimes no difference).
• The variance of a random variable captures the spread of the probability distribution about its mean value. It is defined as

\operatorname{Var}(X)=E\left[(X-\mu)^{2}\right]

and \operatorname{Var}(X) represents the variance of X.

• \operatorname{Var}(X) can also be denoted as the Greek letter \sigma^{2} (called sigma square).
• It is also considered as the long-run average value of the square of the distance from X. Also, notice that there is always \operatorname{Var}(X) \geq 0.
• Instead of having \operatorname{Var}(X)=E\left[(X-\mu)^{2}\right], we can also have
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## 2.2 Expected Value (Mean)

#### Expected Value (Mean)

• You already know what an average is. Expected value is similar in this sense (after all, it is known as mean too).
• Since random variables have a list or range of possible outcomes with probabilities associated, we can determine the mean of the random variable, known as its expected value.
• The formula for expected value is given by

E(X)=\sum_{x} x \cdot \operatorname{Pr}(X=x)=\sum_{x} x \cdot p(x)

where E(X) represents the expected value of X.

• E(X) can also be denoted as the Greek letter \mu (mu), which means the mean of X.
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