Probability Distributions

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• HW1 posted, due Sunday 11:59 PM

Warm-up Questions (Wooclap on Canvas > Assignments)

Random Variables

Random variables are variables that can take different values, and the value they take depends on a probabilistic process.

Examples:

Probability Distributions

A probability distribution is a function that maps a possible outcome of a random variable to its probability of occurrence.

Key Components of Probability Distributions

• Support: values that the distribution can take

• Parameters: values that characterize the distribution

• PDF/PMF: \(f(x)\) , function whose output is a probability of a particular value or range of values

• CDF: \(F(x)\) , function whose output is a cumulative probability up to a particular value

• Expected value/mean: \(\mathbf{E}(X)\)

• Variance: \(\mathbf{E}[(X-\mu)^2]\)

Example

• Is this a valid probability distribution?

• Is this a discrete or continuous distribution?

• What is the support?

• What is \(P(X=3)\)?

• What is \(F(3)\)?

• What is \(\mathbf{E}(X)\)?

• What is \(Var(X)\)?

Example answers

• Is this a valid probability distribution?

• Is this a discrete or continuous distribution?

• What is the support?

• What is \(P(X=3)\)?

• What is \(F(3)\)?

Example answers

• What is \(\mathbf{E}(X)\)?

   

• What is \(Var(X)\)?

  • \(Var(X)=\mathbf{E}[(X-\mu)^2]=\mathbf{E}[X^2]-\mathbf{E}[X]^2\) (do this on your own)

Common Probability Distributions

1. Normal
2. t
3. Bernoulli
4. Binomial
5. Uniform
6. Multinomial
7. Poisson
8. Negative Binomial