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

- Bernoulli
- Binomial
- Negative Binomial
- Poisson

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

- Bernoulli
- Binomial
- Negative Binomial
- Poisson

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

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

Outcome | P |
---|---|

1 | \(p\) |

0 | \(1-p\) |

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 |

In general,

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

In general,

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

In general,

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

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, \[P(\text{Outcome} = x) = 0.\]

Rather than individual outcomes, the focus will be on

**intervals**of outcomes.A key interval of interest is \((-\infty, x]\).

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

\[F_X(x) = P(X \leq x)\]

\[F_X(x) = P(X \leq x)\]

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

Properties

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

- What can we learn from the CDF?

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

- 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?

- 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?

- 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?

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

- 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.

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

- What should the y-axis be labeled?

- Uniform
- Normal
- Exponential
- Gamma
- Beta