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# Variance

## 4.2 Distribution of Sample Proportion

#### Sampling Distribution of a Small Proportion

• Recall that

\text { Sample proportion, }\ \hat{p}=\frac{\text { number in population with attribute }}{\text { population size }}

and can be treated as a random variable.

• In a small enough population, we are able to list out all the possible samples, the probability of getting each sample, and therefore the sample proportion.
• Therefore, we are able to construct the probability distribution of \hat{p}.
• The distribution of a statistic which is calculated from a sample (such as the sample proportion) is called a sampling distribution.
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## 3.2 Mean, Variance, Standard Deviation, Interquartile Range

#### Expected Value (Mean)

• The expected value for a continuous random variable X is given by

E(X)=\int_{-\infty}^{+\infty} x \cdot f(x) d x=\int_{c}^{d} x \cdot f(x) d x

where [c,\ d] is the range of x where, outside of this range, we have f(x)=0.

• As usual E(X) can also be denoted as the Greek letter \mu (mu).
• Expanding the formula, we have
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## 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.
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## Continuous Random Variables

This tutorial covers material encountered in chapter 15 of the VCE Mathematical Methods Textbook, namely: Continuous random variables Probability density functions Mean, variance and standard… Read More »Continuous Random Variables

## The Binomial Distribution

This tutorial covers material encountered in chapter 14 of the VCE Mathematical Methods Textbook, namely: Bernoulli distribution Binomial distribution Mean and variance of a binomial… Read More »The Binomial Distribution

## Introduction to Probability

This tutorial covers material encountered in chapter 13 of the VCE Mathematical Methods Textbook, namely: Basic set and probability theory Addition and multiplication (Bayes’ Theorem)… Read More »Introduction to Probability