Chapter 6: Random Variables

how to find binomial probabilities

*STEP 1*: State the distribution and the values of interest. Specify a binomial distribution with the number of trials n, success probability p, and the values of the variable clearly identified. *STEP 2*: Perform calculations - show your work! Do one of the following: (i) Use the binomial probability formula, or (ii) Use binompdf or binomcdf and label the inputs. *STEP 3*: Answer the question.

standard deviation of a random variable

- A measure of how much the values of the variable typically vary from the mean µx.

expected value of a discrete random variable

- Describes the long-run average outcome of a chance process. - To find µx = E(x), multiply the probability of each outcome by the value of the outcome. Add all possible products.

Normal approximation for binomial distributions

--> As n gets larger, the binomial distribution gets close to a Normal distribution. --> *When do we use a Normal approximation?* When n is so large that np ≥ 10 and nq ≥ 10.

effects of a linear transformation on a random variable

-> If y = a + bx is a linear transformation of the random variable X, then --> the probability distribution of Y has the same shape as the probability distribution of X if b > 0. • µy = a + µx ∂y = |b|∂x REMEMBER: Whether we're dealing with data or random variables (both discrete and continuous), the effects of a linear transformation are the same.

how can I tell if a probability distribution is legitimate?

1) All the probabilities of x add up to 1. 2) All probabilities are between 0 and 1

steps to solve Statistics problems

1) State your distribution and your values of interest. 2) Perform calculations - show your work! 3) Answer the question

binomial setting

A *binomial setting* arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a binomial setting are: *[BINS]* *(1) BINARY*: Two outcomes: 'success' vs 'failure' *(2) INDEPENDENT*: Knowing the result of one trial must not tell us anything about the result of any other trial. *(3) NUMBER*: The number of trials 'n' of the chance process *must be fixed in advance*. *(4) SUCCESS*: There is the same probability p of success on each trial. - In a binomial setting, we can define a random variable (say *X*) as the number of successes in 'n' independent trials.

random variable

A variable that takes numerical values that describe the outcomes of some chance process. The *probability distribution* of a random variable gives its possible values and their probabilities.

discrete random variable

A variable x that takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable x lists the values xi and their probabilities pi (where pi is a number between 0 and 1 and the sum of pi = 1)

continuous random variable

A variable x that takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of x that make up the event. All continuous probability models assign probability to every individual outcome.

effect on a random variable of adding (or subtracting) a constant

Adding the same positive number 'a' to (subtracting 'a' from) each value of a random variable: • Adds 'a' to (subtracts 'a' from) measures of center and location (mean, median, quartiles, percentiles). • Does not change shape or measures of spread (range, IQR, standard deviation).

binomial distributions in statistical sampling

Almost all real-world sampling, such as taking an SRS from a population of interest, is done without replacement. This leads to a violation of the independence condition for a binomial setting.

variance of the difference of random variables

For any two independent random variables X and Y, if D = X - Y, then the variance of D is (∂^2)(D) = (∂^2)(X) + (∂^2)(Y) *Notice you don't subtract the variances. You still add them.*

variance of the sum of independent random variables

For any two independent random variables X and Y, if T = X + Y, then the variance of T is (∂^2)(T) + (∂^2)(Y) *REMEMBER*: (1) You can add variances only if the two random variables are independent. (2) You can never add standard deviations.

mean of the difference of random variables

For any two random variables X and Y, if D = X - Y, then the mean of D is µD = µX - µY. (Remember: The order of subtraction is important)

mean of the sum of random variables

For any two random variables X and Y, if T = X + Y, then the expected value of T is E(T) = µT + µx + µy --> In general, the mean of the sum of several random variables is the sum of their means. --> *REMEMBER*: We can't calculate the probability of any value of T unless X and Y are *independent random variables*.

geometric probability formula

If Y has the geometric distribution with probability p of success on each trial, the possible values of Y are 1, 2, 3... If k is any one of these values, P(Y = k) --> *TIP*: Can be done on the calculator: geometpdf (p, k) computes P(Y = K) geometcdf (p, k) computes P (Y ≤ K)

mean (expected value) of a geometric random variable

If Y is a geometric random variable with probability of success p on each trial, then its *mean* (expected values) is µy = E(y) = 1/p (That is, the expected number of trials required to get the first success is 1/p)

mean and standard deviation of a binomial distribution

If a count X of successes has the binomial distribution with number of trials n and probability of success p, the *mean* and *standard deviation* of x are: *REMEMBER*: These formulas only work for binomial distributions.

independent random variable

If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are *independent random variables*.

large counts condition for a binomial distribution

Suppose that count X of successes has the binomial distribution with n trials and success probability p. When n is large, the distribution of X is approximately Normal with: mean: µx = np and standard deviation: ∂x = √npq *(We will use the Normal approximation when n is so large that np ≥ 10 and nq ≥ 10.)*

binomial random variable

The count X of successes in a binomial setting.

geometric random variable

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

binomial coefficient

The number of ways of arranging k successes among n observations. If X has the binomial distribution with n trials and probability p of success on each trial, the possible values of X are 0, 1, 2, .... n. If k is any of these values, P(X = K) is given by the following formula: *TIP*: You can use binompdf and binomcdf on the calculator to perform this calculation: binompdf (n, p, k) computes *P(X = K)* binomcdf (n, p, k) computes *P(X ≤ K)*

binomial distribution

The probability distribution of a binomial random variable. It has parameters n and p, where *n* is the number of trials of the chance process and *p* is the probability of a success on any one trial. The possible values of X are the whole numbers from 0 to n.

geometric distribution

The probability distribution of a geometric random variable with parameter p, the probability of success on any trial. The possible values of Y are 1, 2, 3....

10% condition for a binomial distribution

When taking an SRS of size n from a population of size N, we can use a binomial distribution to model the count of successes in the sample as long as n ≤ (1/10)N

geometric setting

When we perform independent trials of the same chance 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.

effects on a random variable of multiplying (or dividing) a constant

multiplying (or dividing) each value of a random variable by a positive number: • Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b. • Multiplies (divides) measures of spread (range, IQR, standard deviation) by b. • Does not change the shape of the distribution. --> As with data, if we multiply a random variable by negative constant b, our common measures of spread are multiplied by |B| --> Also, how does multiplying by a constant affect the variance? --------> Multiplying a random variable by a constant 'b' multiplies the variance by b^2

combining Normal random variables

• If a random variable is Normally distributed, we can use its mean and standard deviation to compute probabilities. • Any sum or difference of independent Normal random variables is also Normally distributed. (The µ and ∂ of the resulting Normal distribution can be found using the appropriate rules for means and variances).