Probability and Discrete/Continuous Distributions: Exam 2 Study Notes

Ch4. Probability

Event, Sample Space, Probability

Probability theory is the foundation of statistical inference, describing the likelihood of events within a defined sample space.

• Event: A subset of outcomes from a sample space.

• Sample Space: The set of all possible outcomes.

• Probability: A measure between 0 and 1 representing the chance of an event occurring.

Complementary Events

• Definition: For event A, the complement is the event that A does not occur.

• Formula: $P(A^c) = 1 - P(A)$

Compound Events

• At least A, B, or both: Probability that at least one of A or B occurs.

• Mutually Exclusive Events: Events that cannot occur together (disjoint events).

• Sample with Replacement (Independent): Each draw does not affect the next.

• Sample without Replacement (Dependent): Each draw affects the next.

• Formulas:

  • $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

  • If A and B are mutually exclusive: $P(A \text{ or } B) = P(A) + P(B)$

  • If A and B are independent: $P(A \text{ and } B) = P(A) \times P(B)$

Conditional Probability

• Definition: Probability of A given that B has already happened.

• Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Application: Used to find $P(A \text{ and } B)$ and $P(A|B)$.

Counting: Multiplication of Steps

Counting principles are used to determine the number of ways events can occur.

• Rearrange n items (order matters): $n! = n \times (n-1) \times (n-2) \times \ldots \times 1$

• Rearrange n items with $n_1, n_2, \ldots, n_k$ items alike: $\frac{n!}{n_1! n_2! \ldots n_k!}$

• Permute r out of n (order matters): $nP_r = \frac{n!}{(n-r)!}$

• Choose r out of n (combination, order does not matter): $nC_r = \frac{n!}{r!(n-r)!}$

Ch5. Discrete Distributions

Valid Random Variables

Discrete random variables take on numerical values with associated probabilities.

• Probability function: $p(x) \geq 0$, $\sum p(x) = 1$

• Mean: $\mu = \sum x \cdot P(x)$

• Variance: $\sigma^2 = \sum (x - \mu)^2 P(x)$

Binomial Distribution

• Definition: Number of successes in n independent Bernoulli trials.

• Probability formula: $P(x) = nC_x \cdot p^x \cdot (1-p)^{n-x}$

• Mean: $\mu = np$

• Variance: $\sigma^2 = np(1-p)$

• Example: Flipping a coin n times and counting heads.

Poisson Distribution

• Definition: Number of occurrences in a unit interval.

• Formula: $P(x) = \frac{e^{-\mu} \mu^x}{x!}$

• Mean and Variance: Both equal $\mu$

• Example: Number of emails received per hour.

Ch 6.1, 6.2 Continuous Distributions

Valid Density Curve

Continuous random variables are described by density curves, where the area under the curve represents probability.

• Properties: $f(x) \geq 0$, total area under curve = 1.

• Probability at an exact value: $P(x = a) = 0$

• Uniform Distribution: Probability is constant over the interval; area is a rectangle.

Standard Normal Distribution (N(0,1))

• Definition: Normal distribution with mean 0 and variance 1.

• Use z table: Table gives probability $P(Z < z)$.

• Probability calculations:

  • $P(a < Z < b) = P(Z < b) - P(Z < a)$

  • $P(Z > a) = 1 - P(Z < a)$

• Important critical values:

  • $Z_{0.05} = 1.645$

  • $Z_{0.025} = 1.96$

  • $Z_{0.01} = 2.326$

Regular Normal Distribution (N(μ, σ²))

• Standardize formula: $z = \frac{x - \mu}{\sigma}$

• Convert back to x: $x = z \sigma + \mu$

• Application: Used to find probabilities for any normal distribution by converting to standard normal.