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Introductory Statistics: Probability and Distributions

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Termini in questo insieme (23)

  • What is probability?

    Probability is a numerical measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).

  • Define experiment, sample space, and event in probability.

    An experiment is a process with uncertain outcomes. The sample space (S) is all possible outcomes. An event is a subset of the sample space.

  • Basic probability formula for event A

    P(A) = Number of outcomes in A / Total number of outcomes in S.

  • What are the basic rules of probability?

    For any event A, 0 ≤ P(A) ≤ 1. For the sample space S, P(S) = 1.

  • What is the intersection of two events A and B?

    The intersection (A ∩ B) is the event where both A and B occur, containing outcomes common to both.

  • What is the union of two events A and B?

    The union (A ∪ B) is the event where either A or B or both occur, containing all outcomes in A or B.

  • Define disjoint (mutually exclusive) events.

    Two events are disjoint if they cannot occur simultaneously, so P(A ∩ B) = 0.

  • State the additive rule of probability.

    P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For disjoint events, P(A ∪ B) = P(A) + P(B).

  • What is the complement of an event A?

    The complement (Ac) includes all outcomes in S not in A, with P(Ac) = 1 - P(A).

  • Explain conditional probability P(A|B).

    P(A|B) is the probability of event A occurring given that event B has occurred, calculated as P(A ∩ B) / P(B).

  • State the multiplication rule for probability.

    P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B), the probability both A and B occur.

  • When are two events independent?

    Events A and B are independent if P(A|B) = P(A), equivalently P(A ∩ B) = P(A) × P(B).

  • Additive rule for independent events

    For independent events, P(A ∪ B) = P(A) + P(B) - P(A) × P(B).

  • What is a random variable?

    A random variable assigns a numerical value to each outcome of a random experiment, denoted by a capital letter like X.

  • What is a probability distribution?

    A probability distribution lists all possible values of a random variable and their associated probabilities.

  • Conditions for a legitimate probability distribution

    All probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

  • Define expected value (mean) of a random variable

    The expected value is the long-run average outcome, calculated as the sum of each value times its probability.

  • What is the Empirical Rule?

    For approximately normal data: ~68% within 1 SD, ~95% within 2 SDs, and ~99.7% within 3 SDs of the mean.

  • Describe the normal distribution

    The normal distribution is symmetric, bell-shaped, defined by mean µ and standard deviation σ, and models many natural phenomena.

  • What is the standard normal distribution?

    A normal distribution with mean 0 and standard deviation 1, denoted by Z.

  • How to calculate probabilities using StatCrunch for normal distributions?

    Use StatCrunch's Normal Calculator with known mean and SD to find area under the curve for intervals.

  • What is an inverse normal probability?

    Finding the value of a random variable corresponding to a given cumulative probability in a normal distribution.

  • Law of Total Probability formula

    P(A) = P(B) × P(A|B) + P(Bc) × P(A|Bc), combining probabilities over different pathways.