Unit Four: Probability, Random Variables, and Probability Distributions- essential knowledge

Do patterns in data necessarily mean that variation is not random?

No, patterns in data do not necessarily mean that variation is not random.

What determines the results generated by a random process?

A random process generates results that are determined by chance.

What is an outcome in the context of a random process?

An outcome is the result of a trial of a random process.

What is an event in probability?

An event is a collection of outcomes.

How is simulation used in probability?

Simulation models random events so that simulated outcomes closely match real-world outcomes.

What is recorded during a simulation to estimate probabilities?

The counts of simulated outcomes and the total count are recorded.

How can the relative frequency of an outcome be used in probability?

The relative frequency can be used to estimate the probability of an outcome or event.

What does the law of large numbers state?

It states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increases.

What is the sample space of a random process?

The sample space is the set of all possible non-overlapping outcomes.

How is the probability of an event calculated if all outcomes are equally likely?

Probability = (Number of outcomes in event E) / (Total number of outcomes in the sample space).

What range does the probability of an event fall within?

The probability is between 0 and 1, inclusive.

How do you find the probability of the complement of an event E?

The probability of the complement of E is 1 – P(E).

How can probabilities in repeatable situations be interpreted?

They can be interpreted as the relative frequency with which the event will occur in the long run.

What is the joint probability of events A and B?

The joint probability is the probability of the intersection of A and B, denoted P(A ∩ B).

When are two events considered mutually exclusive or disjoint?

When they cannot occur at the same time, so P(A ∩ B) = 0.

What is conditional probability?

Conditional probability is the probability that event A will occur given that event B has occurred, denoted P(A | B).

What does the multiplication rule state?

The probability that events A and B both occur is P(A ∩ B) = P(A) P(B | A).

When are two events A and B considered independent?

They are independent if knowing whether A has occurred does not change the probability that B will occur.

What is the condition for independence in probability?

If events A and B are independent, then P(A | B) = P(A), P(B | A) = P(B), and P(A ∩ B) = P(A) ⋅ P(B).

How do you calculate the probability that event A or event B (or both) will occur?

Use the addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

What are the values of a random variable?

The values are numerical outcomes of random behavior.

What is a discrete random variable?

A variable that can only take a countable number of values, each with a probability, where the sum of all probabilities is 1.

How can a probability distribution be represented?

As a graph, table, or function showing the probabilities associated with values of a random variable.

What does a cumulative probability distribution show?

It shows the probability of being less than or equal to each value of the random variable.

What information does an interpretation of a probability distribution provide?

It provides information about the shape, center, and spread of a population, and allows conclusions about the population of interest.

What is a parameter in the context of a population or distribution?

A parameter is a single, fixed numerical value measuring a characteristic of the population or distribution.

How do you calculate the mean (expected value) for a discrete random variable X?

The mean is calculated as μ = Σ[xiP(xi)].

How do you calculate the standard deviation for a discrete random variable X?

The standard deviation is calculated as σ = √Σ[(xi - μ)^2P(xi)].

How should parameters for a discrete random variable be interpreted?

Using appropriate units and within the context of a specific population.

For random variables X and Y, what is the mean of aX + bY?

The mean is aμx + bμy.

When are two random variables independent?

When knowing information about one does not change the probability distribution of the other.

For independent random variables X and Y, how do you calculate the mean and variance of aX + bY?

The mean is aμx + bμy, and the variance is a^2σx^2 + b^2σy^2.

What happens to the probability distribution when a linear transformation Y = a + bX is applied?

The shape remains the same if a > 0 and b > 0. The mean becomes μy = a + bμx, and the standard deviation becomes σy = bσx.

How can a probability distribution be constructed?

It can be constructed using the rules of probability or estimated with a simulation using random number generators.

What is a binomial random variable?

A variable that counts the number of successes in n independent trials, where each trial has two possible outcomes (success or failure) with a probability of success p.

How do you calculate the probability that a binomial random variable X has exactly x successes?

Use the binomial probability function: P(X = x) = (n choose x)p^x(1 - p)^(n - x).

What are the mean and standard deviation of a binomial random variable?

The mean is μx = np and the standard deviation is σx = √[np(1 - p)].

How should probabilities and parameters for a binomial distribution be interpreted?

Using appropriate units and within the context of a specific population or situation.

What is a geometric random variable?

A variable that gives the number of the trial on which the first success occurs, with each trial having two possible outcomes (success or failure) and a probability of success p.

How do you calculate the probability that the first success occurs on trial x in a geometric distribution?

Use the geometric probability function: P(X = x) = (1 - p)^(x - 1)p.

What are the mean and standard deviation of a geometric random variable?

The mean is μx = 1/p and the standard deviation is σx = √[(1 - p)/p^2].

How should probabilities and parameters for a geometric distribution be interpreted?

Using appropriate units and within the context of a specific population or situation.