## Review

Up until now, we have focused primarily on discrete distributions:

• Bernoulli
• Binomial
• Negative Binomial
• Poisson

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

### Bernoulli

Outcome P
1 $$p$$
0 $$1-p$$

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

Outcome P
0 0.237
1 0.396
2 0.264
3 0.088
4 0.015
5 0.001

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

### Binomial(N, p)

In general,

$$P(Outcome = x) = {N \choose x}p^x(1-p)^{N-x}$$

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

### Negative binomial(k, p)

In general,

$$P(Outcome = x) = {k-1 \choose x}p^k(1-p)^{x}$$

## Review

Discrete probability distributions have the property that we can calculate a probability for each possible outcome.

### Poisson(λ)

In general,

$$P(Outcome = x) = \frac{\lambda^xe^{-\lambda}}{x!}$$

## Continuous Models

In contrast to discrete probability models where $P(\text{Outcome} = x) > 0$ for at least one candidate outcome, the outcomes of continuous probability models all have zero probability, $P(\text{Outcome} = x) = 0.$

## Continuous Models

• Rather than individual outcomes, the focus will be on intervals of outcomes.

• A key interval of interest is $$(-\infty, x]$$.

• $$P(X\in (-\infty, x]) = P(X \leq x)$$

## The Cumulative Distribution Function

$F_X(x) = P(X \leq x)$

## The Cumulative Distribution Function

$F_X(x) = P(X \leq x)$

• Note that the CDF is also applicable to discrete probability models.

## The Cumulative Distribution Function ## The Cumulative Distribution Function

Properties

• Ranges from 0 to 1
• Monotonically increasing (non-decreasing)

## Mystery CDF ## Mystery CDF

• What can we learn from the CDF?

## Mystery CDF

• What can we learn from the CDF?
• What is going on in the regions that are flatter?

## Mystery CDF

• What can we learn from the CDF?
• What is going on in the regions that are flatter?
• What is going on in the regions that are steeper?

## Mystery CDF

• What can we learn from the CDF?
• What is going on in the regions that are flatter?
• What is going on in the regions that are steeper?
• Where is the median?

## Mystery CDF

• What can we learn from the CDF?
• What is going on in the regions that are flatter?
• What is going on in the regions that are steeper?
• Where is the median?
• Where is the 25th percentile? 75th percentile?

## The slope and relative likelihood

• Values in flatter intervals are less likely to occur than values in steeper intervals.

## The slope and relative likelihood

• Values in flatter intervals are less likely to occur than values in steeper intervals.
• The derivative of the CDF (if it exists) is the relative likelihood, also called the probability density function.

$f_X(x) = \frac{d}{dx} F_X(x)$

## Probability density function ## Probability density function

• What should the y-axis be labeled?

• Uniform
• Normal
• Exponential
• Gamma
• Beta