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What is a random variable?
If the outcome of a probability experiment is a numerical result, we say that the outcome is a random variable.
Random variables are typically denoted using capital letters such as "X".
We follow the practice of using a capital letter, such as "X", to identify the random variable and a lowercase letter, "x", to list the possible values of the random variable, or the sample space of the experiment.

What are the two types of random variable? Define each.
A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point.
Discrete random variables typically result from counting (e.g., # of people in line at a bank).
A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion.
Typically result from measurement (e.g., measuring time between calls to a service center).

Define probability distribution.
The probability distribution of a discrete random variable "X" provides (1) the possible values of the random variable and (2) their corresponding probabilities. A probability distribution can be in the form of a table, graph, or mathematical formula.
In simpler terms, because the value of a random variable is determined by chance, we may assign probabilities to the possible values of the random variable.

How many rules are there for a discrete probability distribution? Give this/these rules.
Let P(x) denote the probability that the random variable "X" equals "x"; then
1) The sum of the probabilities must equal one.
2) All probabilities must be between zero and one inclusive.

In the graph of a discrete probability distribution, the __________ axis is the _____ of the ________ ______ ________ and the ________ axis is the _____________ ___________ of the discrete random variable. When graphing a discrete probability distribution, we want to emphasize that the data are ________. Therefore, draw the graph of discrete probability distributions using ________ lines _____ each value of the random variable to a ______ that is the probability of the random variable.
horizontal, value, discrete random variable, vertical, corresponding probability, discrete, vertical, above, height

Give the formula for the mean of a discrete random variable?
We round the mean to one more decimal place than the value of the random variable.

As the ______ of repetitions of the experiments increases, the ____ value of the ___ trials will approach __.
number, mean, "n", (the mean for the random variable "X")

As the number of repetitions of the experiments increases, the __________ between ___ and __ gets ______ to ____.
difference, , , closer, zero

Because the mean of a ______ ________ represents what we would ______ to happen in the ____ ___, it is also called the ________ _____, __. The interpretation of the expected value is ___ ____ __ the interpretation of the ____ of a ________ ______ ________.
random variable, expect, long run, expected value, E(X), the same as, mean, discrete random variable
E(X) is read as follows: the expected value of the random variable "x". When you hear "expected value of a random variable", think "mean of the random variable".

Give the formula for computing the standard deviation of a discrete random variable.

Finding the Variance of a Discrete Random Variable:
The ________ of the discrete random variable, ___, is the value under the ______ ____ in the computation of the ________ _________.
variance, , square root, standard deviation

6.2
What are the four criteria for a binomial experiment?
(1) The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial.
(2) The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials.
(3) For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure.
(4) The probability of success is fixed for each trial of the experiment.

6.2
T or F
If asked to verify that an experiment is a probability experiment, you need to verify that all four criterion for a binomial experiment are satisfied.
True

6.2
What do , , and represent when working with binomial probability distribution?
n = the number of independent trials of the experiment
p = the probability of success
1p = the probability of failure
X (capital) = binomial random variable that denotes the number of successes in "n" independent trials of the experiment.
So,

6.2
If is a ________ ______ ________ that denotes the ______ of _________ in ___________ trials of an __________, the possible values of are _____.
binomial random variable, number, successes, independent, experiment,

6.2
In the binomial probability distribution function, what does represent?
(hint: is the number of ways to get "x" success(es) in "n" trials)
The probability of obtaining "x" success in "n" independent trials of a binomial experiment. This probability is given by
where "p" is the probability of success.

6.2
In the probability distribution function, what does represent?
the probability of success raised to the number of successes

6.2
In the binomial probability distribution function, what does represent?
the probability of failure raised to the "nx" number of failures.

6.2
State the Binomial Probability Distribution Function (pdf, not cdf).
The probability of obtaining "x" successes in "n" independent trials of a binomial experiment is given by
x=0, 1, 2, ..., n where "p" is the probability of success.

6.2
What is "np"?
"np" is the expected number of successes in a binomial experiment with "n" trials and probability of success "p".

6.2
match the symbol with the phrase:
more than or greater than
no more than or at most or less than or equal to
exactly or equals or is
fewer than or less than
at least or no less than or greater than or equal to
more than or greater than ( )
no more than or at most or less than or equal to ( )
exactly or equals or is (=)
fewer than or less than (<)
at least or no less than or greater than or equal to ( )

6.2
State the formulas for the mean (or expected value) and standard deviation of a binomial random variable.
The ____ of a ________ ______ ________ ______ the _______ of the ______ of trials, "n", of the experiment and the ___________ of _______, p. It can be ___________ as the ________ ______ of _________ after "n" trials of the __________.
mean = =
standard deviation =
mean, binomial random variable equals, product, number, probability, success, interpreted, expected number, successes, experiment

6.2
To graph a ________ ___________ ____________, first find the ___________ for each ________ _____ of the random variable. Then follow the ____ approach as was use to graph ________ ___________ _____________.
binomial probability distribution, probabilities, possible value, same, discrete probability distributions

6.2
T or F
If p < 0.5, the binomial probability distribution is skewed left.
If p = 0.5, the binomial probability distribution is symmetric.
If p > 0.5, the binomial probability distribution is skewed right.
False,
If p < 0.5, the binomial probability distribution is skewed right.
If p = 0.5, the binomial probability distribution is symmetric.
If p > 0.5, the binomial probability distribution is skewed left.

6.2
T or F
As "n" increases, the shape of a binomial probability distribution is leftskewed as the number of success, p, increases.
T or F
As "n" increases, the shape of a binomial probability distribution is rightskewed as the number of success, p, decreases.
T or F
If the value of "p" is < 0.5 and is held constant, as "n" increases, the shape of the distribution is bellshaped.
T or F
If the value of "p" is > 0.5 and is held constant, as "n" increases, the shape of the distribution is rightskewed.
T or F
If the value of "p" is = 0.5 and is held constant, as "n" increases or decreases, the shape of the distribution remains bellshaped.
False, ...is right skewed as the number of success, p, increases.
False, ...is leftskewed as the number of success, p, decreases.
False, ...has a bellshape in the midst of a distribution that may be symmetric at one point but becomes skewedright the more "n" increases.
False, ...has a bellshape in the midst of a distribution that may be symmetric at one point but becomes skewedleft the more "n" increases.
True

6.2
Under what conditions will a binomial probability distribution be approximately bellshaped?
If/When p = 0.5, for a fixed "p", as the number of trials "n" in a binomial experiment increases, the probability distribution of the random variable "X" becomes bellshaped.
If np(1p) 10, the probability distribution will be approximately bell shaped.

6.2
Explain how to determine if an observation in a binomial experiment is unusual.
If the observation is less than or greater than , it is unusual according to the Empirical Rule.

6.2
TI84 tips:
Use "binompdf(" when "exactly or equals or is" is stated
Use "binomcdf(" otherwise.
How to use "binomcdf":
"no more than" = binomcdf(  no extra math
"more than" = binomcdf(  subtract outcome from 1
"between any two variables" = binomcdf(  subract on less than the lesser binomcdf( value from greater value (this means calculate for the greater value first, then subtract the lesser outcome from that result).

