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.

Bernoulli

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

Binomial(N = 5, p = .25)

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

Binomial(N, p)

In general,

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

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

Negative binomial(k, p)

In general,

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

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

Poisson(λ)

In general,

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

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