AP Stats Ch6

Random Variable

A(n) ____ takes numerical values that describe the outcomes of a random process.

Probability Distribution

The ____ of a random variable gives its possible values and their probabilities.

Discrete Random Variable

A(n) ____ X takes a fixed set of possible values with gaps between them.

Mean (expected value) of a discrete random variable

A(n) ____ is its average value over many, many trials of the same random process. Suppose that X is a discrete random variable with probability distribution:

Standard deviation of a discrete random variable

The ____ measures how much the values of the variable typically vary from the mean in many, many trials of the random process. Suppose that X is a discrete random variable with probability distribution:

Variance

\____ is standard deviation of a discrete random variable squared.

Continuous random variable

A(n) ____ can take any value in an interval on the number line.

Adding or Subtracting

The effect of ____ a constant on a probability distribution: ____ the constant to the measures of center and locationDoes not change measures of variabilityDoes not change the shape of the probability distribution

Multiplying or Dividing

The effect of ____ a constant on a probability distribution: ____ the measures of center and location by the constant ____ the measures of variability by the constantDoes not change the shape of the probability distribution

Linear transformation

The effect of A(n) ____ on a random variable:If Y = a + bX is a ____ of the random variable X: -The probability distribution of Y has the same shape as the probability distribution of X if B>0-μ(y) = a + b\*μ(x)-σ(y) = |b|σ(x) (because b could be a negative number).

Mean (expected value) of a sum of random variables

The ____:For any two random variables X and Y, if S = X + Y, the mean (expected value) of S is μ(s) = μ(x+y) = μ(x) + μ(y). In other words the mean is equal to the sum of their means.

Mean (expected value) of a difference of random variables

The ____:For any two random variables X and Y, if D = X - Y, the mean (expected value) of D is μ(d) = μ(x-y) = μ(x) - μ(y). In other words the mean is equal to the difference of their means.

Independent random variables

If knowing the value of X does not help us predict the value of Y, then X and Y are ____. In other words, two random variables are _____ if knowing the value of one variable does not change the probability distribution of the other variable.

Standard Deviation of the sum of two independent random variables

The ____:For any two independent random variables X and Y, if S = X + Y, the variance of S is σ(S)^2 = σ(x+y)^2 = σ(x)^2 + σ(y)^2.To get the standard deviation of S, take the square root of the variance.

Standard Deviation of the difference of two independent random variables

The ____:For any two independent random variables X and Y, if D = X - Y, the variance of D is σ(D)^2 = σ(x-y)^2 = σ(x)^2 + σ(y)^2.To get the standard deviation of D, take the square root of the variance.

Binomial Setting

A ____ arises when we preform n independent trials of the same random process and count the number of times that particular outcome (called a success) occurs.

BINS

Binary: The possible outcomes of each trial can be classified as "success" or "failure"Independent: Trials must be independent.Number: The number of trials n of the random process must be fixed in advance.Same probability: There is the same probability of success p on each trial.

Binomial Random Variable

The count of success X in a binomial setting is a ____.

Binomial Distribution

The probability distribution of X in a binomial setting is a ____. Any ____ is completely specified by two numbers: the number of trials n of the random process and the probability of p of success on each trial.

Binomial Coefficient

The number of ways to arrange x successes among n trials is given by the ____.nCr = n! / x! * (n - x)!

Binomial Probability Formula

\____: Suppose that X is a binomial random variable with n trials and probability p of success on each trial. The probability of getting exactly x successes in n trials (x = 0, 1, 2, ..., n) is;P(X = x) = nCr * p^x * (1 - p)^n-x

Mean (expected vaule) of a binomial random variable

\____: If a count X of successes has a binomial distribution with number of trials n and probability of success p, the mean (expected value) of X is;μ(x) = E(X) = n * p

Standard Deviation of a binomial random variable

\____: If a count of successes has a binomial distribution with number of trials n and probability of success p, the standard deviation of X is;σ(x) = √np * (1 - p)

10% Condition

\____: When taking a random sample of size n from a population of size N, we can treat individual observations as independent when preforming calculations as long as n < 0.10*N

Large Counts Condition

\____: Suppose that a count X of successes has a binomial distribution with trials n and success probability p. The ____ says that the probability distribution of X is approximately normal if np is greater than or equal to 10 and n(1-p) is greater than or equal to 10. That is, the expected numbers (counts) of successes and failures are both at least 10.

Geometric Setting

A ____ arises when we preform independent trials of the same random process and record the number of trials it takes to get one success. On each trial, the probability p of success must be the same.

BITS

Binary: The possible outcomes of each trial can be classified as "success" or "failure"Independent: Trials must be independent.Trials: How many trials until the first success.Same probability: There is the same probability of success p on each trial.

Geometric Random Variable

The number of trials X that it takes to get a success in a geometric setting is a ____.

Geometric Distribution

The probability distribution of X in a geometric setting is a ____ with probability p of success on any trial. The possible values of X are 1, 2, 3, ....

Geometric Probability Formula

\____: If X has a geometric distribution with probability p of success on each trial, the possible values of X are 1, 2, 3, .... If x is anyone of these values;P(X = x) = p * (1 - p)^x-1

Mean (expected value) of a geometric random variable

If X is a geometric random variable with probability of success p on each trial, then its mean (expected value) is:μ(x) = E(X) = 1 / p

Standard Deviation of a geometric random variable

If X is a geometric random variable with probability of success p on each trial, then its standard deviation is:σ(x) = √1-p / p