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# Expectation and Variance

## 2.5 Properties of the Binomial Distribution

#### Graph of the Binomial Distribution

• Recall that a binomial distribution has the format

\operatorname{Pr}(X=x)=\left(\begin{array}{l}n \\x\end{array}\right) p^{x}(1-p)^{n-x} \quad \ \ \ \ \ \ \ \ \ x=0,1,2, \ldots, n

where X \sim \operatorname{Bi}(n, p).

• We can plot the graph of (number of successes) against \operatorname{Pr}(X=x) \text { or } p(x) (corresponding probabilities) to see the graphical representation of the binomial distribution.
• The graph of the binomial distribution will generally look like a small mountain. In other words, you would expect probability of number of successes to be higher when x is roughly half the number of trials.
• However, it is obvious that if p is higher, then we would expect more successes. Therefore, the value of p will cause the graph to skew sideways. Refer to the example below to see what ‘positively/negatively skewed’ means graphically. To give a heads up, it means the graph would be slanted towards the left/right.
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