6.1, 6.2., 6.3

median of a discrete random variable

-50th percentile of its probability distribution -can find the median from a cumulative probability distribution -the median of a discrete random variable is the smallest value for which the cumulative probability equals or exceeds 0.5term-8

combining normal random variables

-any sum or difference of independent Normal random variables is also Normally distributed (any linear combination of independent Normal random variables is Normally distributed) -the mean and standard deviation of the resulting Normal distribution can be found using the appropriate rules for means and standard deviations

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 values of variable clearly identified. Step 2: Perform calculations-show your work! -use the binomial probability formula to find the desired probability -or use the binompdf or binomcdf command and label each of the inputs (Be sure to answer the question that was asked)

binomial probability formula

Suppose that X is a binomial random variable with "n" trials and probability "p" of successes on each trial. The probability of getting exactly "x" successes in "n" trials (x=0, 1, 2, ..., n) -P(X=x)=(binomial coefficient)(probability of x)(1-probability raised to the n-x power)

the effect of a linear transformation on a random variable

If Y=a+bX is a linear transformation of the random variable X, -the probability distribution of Y has the same shape as the probability distribution of X if b>0 -mean of a linear transformation=a+b(mean) -the standard deviation of a linear transformation=(absolute values of b) multiplied by the standard deviation (b could be a negative number) -results apply to both discrete and continuous random variables

standard deviation 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 standard deviation of X is -the standard deviation=the square root of (trials)(probability)(1-probability)

mean (expected value) of a binomial random variable

If a count X successes has a binomial distribution with number of trials "n" and probability of success "p," the mean (expected value) of X is -mean=the number of trials multiplied by the probability

standard deviation of the difference of two independent random variables

for any two independent random variables X and Y, if D=X-Y, the variance of D is -the standard deviation of D squared=the standard deviation of X squared+the standard deviation of Y squared -to get the standard deviation of D, take the square root of the variance -results apply to both discrete and continuous random variables

standard deviation of the sum of two independent random variables

for any two independent random variables X and Y, if S=X+Y, the variance is -the standard deviation of S squared=the standard deviation of X squared+the standard deviation of Y squared -to get the standard deviation of S, take the square root of the variance -results apply to both discrete and continuous random variables

geometric probability formula

if X has the geometric distribution with probability "p" of success on each trial, the possible values of X are 1, 2, 3,...if "x" is any of these variables P(X=x)=one minus the probability raised to the x-1 power multiplied by the probability

mean (expected value) and standard deviation of geometric random variable

if X is a geometric random variable with probability of success "p" on each trial, then its mean (expected value) is mean=E(X)=1/p and its standard deviation is standard deviation equals= the square root of 1-minus the probability divided by the probability

independent random variables

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

binomial coefficient

it gives the number of ways to arrange x successes among n trials is given by the binomial coefficient (ex. 6 CHOOSE 4)

mean (expected value) of a discrete random variable

its average values over many, many trials of the same random process; to find the expected value of X, multiply each possible value of X by its probability, and then add all of the products

standard deviation of a discrete random variable

measures how much the values of the variable typically vary from the mean in many, many trials of the random process -value when you take the square root of the variance

the effect of multiplying or dividing by a constant on a probability distribution

multiplying (or dividing) each value of a random variable by the same positive number b: -multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b -multiplies (divides) measures of of variability (range, IQR, standard deviation) by b -does not change the shape of distribution

probability distribution

of a random variable gives its possible values and their probabilities

mean and standard deviation of a linear combination of random variables

of aX+bY is linear combination of the random variables X and Y, -its mean is a(mean of X)+b(mean of Y) -its standard deviation is the square root of the sum of (a squared)(standard deviation of X squared)+(b squared)(standard deviation of Y squared) -results apply to discrete and continuous random variables

Large Counts Condition

suppose the a count of X successes had the binomial distribution with "n" trials and success probability "p." The Large Counts Condition says that the probability distribution of X is approximately Normal if -np is greater than or equal to 10 -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

random variable

takes numerical values that describe the outcomes of a random process ex. number of heads obtained in three coin tosses -capital, italic letters (usually X or Y) are used to designate random variables -lowercase, italic letter (still usually x or y) are used to designate specific values of those variables

binomial random variable

the count of successes X in a binomial setting (The possible values of X is are 0, 1, 2,..., n

geometric random variable

the number of trials X that it takes to get a success is a geometric random variable

binomial distribution

the probability distribution of X; any binomial distribution is completely specified by two numbers: the number of trials "n" of the random process and the probability p of success on each trial

variance of a discrete random variable

the weighted average of squared deviations (weighted average is the mean subtracted from the value and the outcome squared and multiplied by the probability) -the variance is the sum of the mean subtracted from the value and that outcome squared and multiplied by the value's probability for each value in the distribution

geometric distribution

with probability "p" of success on any trial. The possible values of X are 1, 2, 3,...

binomial setting

-arises when we perform "n" independent trials of the same random process and count the number of times that a particular outcome (called a "success") occurs the four conditions for a binomial setting are: -Binary? The possible outcomes of each trial can be classified as "success" or "failure." -Independent? Trials must be independent. That is, knowing the outcome of one trial must not tell us anything about the outcome of any other trial -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

probability distribution for a discrete random variable

-every probability included in the table is a number between 0 and 1, inclusive -the sum of the probabilities is equal to 1

mean (expected value) of a difference of random variables

-for any two random variables X and Y, if D=X-Y, the mean (expected value) of D is -mean of the difference of two random variables X and Y= difference of their means -results apply to both discrete and continuous random variables

mean (expected value) of a sum of random variables

-for any two random variables X and Y, if S=X+Y, the mean (expected value) of S is: -mean of a sum of random variables X and Y=the sum of their means -results apply to both discrete and continuous random variables

mean and standard deviation for continuous random variables

-the mean is the point at which the area under the density curve would balance if it were made of solid material (for a symmetric density curve like a Normal curve, it would be at the exact center of the distribution) -the standard deviation of the Normal distribution is found by observing the inflection points of the curve

how to find probabilities for a continuous random variable

-the probability of any event involving a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that make up the event -all continuous probability distributions assign probability 0 to every individual outcome -the probability distribution for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes

10% condition

When taking a random sample of size "n" from a population of size N, we can treat individual observations as independent when performing calculations as long as n<0.10N

discrete random variable

X takes a fixed set of possible values with gaps between them -typically result from counting something

the effect of adding or subtracting a constant on a probability distribution

adding the same positive number "a" to (subtracting "a" from) measures of center and location (mean, median, quartiles, percentiles) -does not change measures of variability (range, IQR, standard deviation) -does not change the shape of the probability

geometric setting

arises when we perform 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

continuous random variable

can take any value in an interval on the number line