# Mean

## 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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## 2.2 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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## The Normal Distribution [Video Tutorial]

This tutorial covers material encountered in chapter 16 of the VCE Mathematical Methods Textbook, namely: Normal random variables The standard normal distribution Standardising a normal… Read More »The Normal Distribution [Video Tutorial]

## Continuous Random Variables [Video Tutorial]

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 [Video Tutorial]

## The Binomial Distribution [Video Tutorial]

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 [Video Tutorial]

## Introduction to Probability [Video Tutorial]

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 [Video Tutorial]