Chapter 5-7

Probability and Random Variables Flashcards for Exam Review

Law of Large Numbers

If we observe many repetitions of a chance process, the total proportion of times that a specific outcome occurs approaches a single value; this value is the probability of that outcome.

Probability

A number between 0 and 1 that describes the approximate proportion of times an outcome would occur in a very long series of repetitions; often called a long-run relative frequency.

Outcome

One result of a chance process.

Sample Space

The set of all possible outcomes of a chance process.

Event

Any outcome or combination of outcomes from some chance process.

The Complement Rule

The probability of A not occurring is 1 – P(A); P(AC) = 1 – P(A)

The Probability of A and B: P(A ∩ B)

P(A ∩ B) is the “overlap” of events A and B in many situations.

Mutually Exclusive Events and P(A ∩ B)

If A and B are mutually exclusive, they can’t both happen and P(A ∩ B) = 0.

The Probability of A or B: P(A ∪ B)

The probability of A or B occurring is P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

Mutually Exclusive Events - A and B

If events A and B are mutually exclusive, they can’t both happen and P(A ∩ B) = 0.

“Neither” Probability

The probability of neither A nor B happening = 1 – P(either A or B happening); P(neither A nor B) = 1 – P(A ∪ B)

Conditional Probability

The probability that A will happen, given that B has already happened, is P(A | B) = P(A ∩ B) / P(B)

Independent Events

If events A and B are independent, they have no impact on one another; whether or not B occurs has no impact on the probability of A occurring, so P(A | B) = P(A)

The Probability of “At Least One Success”

The probability that event A occurs at least once in n trials is 1 – the probability that it never occurs in n trials; P(at least one success) = 1 – P(AC)n

Random Variable

Takes numerical values that describe the outcome of a chance process.

Probability Distribution of a Random Variable

Gives its possible values and their probabilities.

Discrete Random Variables

Outcomes can only take certain values; each outcome has its own probability between 0 and 1, and the probabilities of all the outcomes add to 1.

Mean µX or Expected Value E(X) of a Discrete Random Variable X

A weighted average of the possible values of X, taking into account the fact that not all values may be equally likely; µ! = 𝐸(𝑋) = 𝑥"𝑝" + 𝑥#𝑝# + ⋯ + 𝑥$𝑝$ = .𝑥%𝑝%

Standard Deviation σX of a Discrete Random Variable X

The approximate average distance between the values of the random variable X and the mean or expected value µX; 𝜎! = 0(𝑥" − µ!)#𝑝" + (𝑥# − µ!)#𝑝# + ⋯ + (𝑥$ − µ!)#𝑝$ = 2.(𝑥% − µ!)#𝑝%

Continuous Random Variable

Takes any value in an interval of values; the probability distribution is described by a density curve; the probability of an event taking place is equal to the area under that region of the curve.

Binomial Setting

Perform several trials of a chance process and record the number of times a particular outcome occurs (or in other words, the count of successes of the trials).

Binomial Random Variable

The count of successes X in a binomial setting.

Binomial Distribution

The probability distribution of X with number of trials n and probability of success on each trial p; the possible values of X are whole numbers from 0 (no successes) to n (success on each trial).

Geometric Setting

Perform independent trials of the same chance process and record the number of trials it takes to get one success.

Geometric Random Variable

The number of trials Y that it takes to get one success in a geometric setting.

Geometric Distribution

The probability distribution of Y, the number of trials it takes to get one success, with parameter p, the probability of success on any trial; the possible values of Y are 1, 2, 3, …

Parameter

A number that describes some characteristic of a population.

Statistic

A number that describes some characteristic of a sample.

Estimator

A sample statistic that is used to estimate a population parameter.

Unbiased Estimator

A sample statistic with a sampling distribution that is centered on the value of the population parameter.

Sampling Variability

A measure of how much the statistic varies from sample to sample; the standard deviation of the sampling distribution is one measure of sampling variability.

Central Limit Theorem (CLT)

Even if the population distribution isn’t normal, the sampling distribution of x̄ will approach normality as the sample size increases.