# Inferential Statistics

## Probability Distributions

• The Normal Distribution
• The t Distribution
• The Binomial Distribution

### Random Variables

Random variable: a numerical characteristic that takes on different values due to chance

• A discrete random variable has a countable set of distinct possible values.
• A continuous random variable is such that any value (to any number of decimal places) within some interval is a possible value.

Probability distribution: A table, graph, or formula that gives the probability of a given outcome's occurrence

### Discrete Probability Distributions

• Binomial Distributions
• Poisson Distributions

### Continuous Probability Distributions

• Normal Distributions
• Uniform Distributions
• Exponential Distributions

### Discrete Probability Distributions

What if we flipped a fair coin four times? What are the possible outcomes and what is the probability of each?

Heads 0 1 2 3 4
Probability 0.0625 0.25 0.375 0.25 0.0625
Cummulative Probability 0.0625 0.3125 0.6875 0.9375 1

$P(X=0)=0.0625 \\ P(X<3)= P(X=0)+P(X=1)+P(x=2)$

A census was conducted at a university. All students were asked how many tattoos they had.

Tattoos 0 1 2 3 4
Probability 0.85 0.12 0.015 0.01 0.005

$$P(X=0)=.85, \quad P(X=1)=.12, \quad P(X=2)=.015,$$ etc.

### Two common discrete distributions

• The Binomial distribution is a discrete probability distribution. It describes the outcome of n independent trials in an experiment. Each trial is assumed to have only two outcomes, either success or failure.

Suppose there are twelve multiple choice questions in an English class quiz (1 of 5 is correct answer). The probability distribution for the number of correct answers when answered randomly.

• The Poisson distribution is the probability distribution of independent event occurrences in an interval.

If there are 12 cars crossing a bridge per minute on average, the probability distribution for the number of cars crossing the bridge in a minute.

### Common continuous distributions

• Uniform Distribution is a continuous distribution that has constant probability (every possible outcome has an equal chance, or likelihood, of occurring).

For eight-week-old babies can smile between 0 and 23 seconds and any smiling time between 0 to 23 is equally likely . The probability distribution for a randomly selected babies' smiling time.

• Normal Distribution is a continuous distribution that follows a commonly observed, natural probability pattern.

The probability distribution for the height of a randomly selected person.

### Probability Density Functions

Probability Mass Functions for Discrete distributions

Probability Density Functions for Continuous Distributions

### Binomial

$$f(x)=\textstyle {n \choose k}\, p^k (1-p)^{n-k} \quad where$$
n= number of trials,
k= number of successes

### Poisson

$$f(x)=\frac{\lambda^k e^{-\lambda}}{k!} \quad where$$
$$\lambda$$ = is the average number of events per interval (event rate)
k= number of occurrences

### Uniform

$f(x)=\begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases}$

### Normal

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$