A numerical variable that describes the outcomes of a chance process- the probability model for this is its probability distribution
of a random variable, this gives the possible values and their probabilities
discrete random variable
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
Value: x1 x2... Prob: P1 p2...
Which 2 requirements must probabilities satisfy? in other words, how can you show that a probability distribution is legitimate?
-every probability Pi is a number between 0 and 1 -the sum of the probabilities is 1
probability of discrete random variables can be modeled by what?
mean (expected value) of a discrete random variable
an average of the possible outcomes, with each outcome weighted by its probability
formula for expected value (mean)
multiply each possible value by its probability then add up all of the products
continuous random variable
takes on all values in an interval of numbers
how is the probability distribution of a continuous random variable described?
by a density curve where the probability of an event is the area under the curve.
how many values can a continuous random variable have?
trick question: it can have and infinite number of values
do intervals of values have positive or negative probability?
what is the effect of multiplying/dividing a random variable by a constant b?
-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
how does multiplying/dividing a random variable by a constant b affect the variance
multiplies/divides it by b^2
what is the effect of adding/subtracting a constant b on a random variable?
-adds b to measures of center and location(mean, median, quartiles, percentiles -does not change measures of spread (range, IQR, standard deviation) -does not change the shape of the distribution
mean of the sum of random variables
for any 2 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
what kind of variables do X and Y have to be in order to determine the probability for any value T
independent random variables - if knowing whether any event involving x alone has occurred tells us nothing about the occurrence of any event involving y alone
if T=X+Y, and x and y are independent random variables, then hoe do you find the variance of T?
varianceT=varianceX+varianceY in general, the variance of the sum of several independent random variables is the sum of their variances *you can only add variances if the 2 are independent and you can NEVER ADD STANDARD DEVIATIONS
For any two random variables x and y, if D=x-y, what is the formula for the expected value of d?
In general, the mean of the difference of several random variables is the difference of their means. The order of subtraction is important
For any two independent random variables x and y, if d=x-y, what is the formula for the variance of D
How is any sum or difference of independent normal random variables distributed
as far as shape, center, and spread, what does the transformation of x to y=a+bx change?
Shape: same as the probability distribution of x Center: meanY=a+b(µX) Spread: standard deviationY=|b|(standard deviation x)
Do x and y have to be independent in order to find the variance of x±y
When does a binomial setting arise?
When we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. Must meet 4 binomial conditions (BINS)
What are the four conditions for a binomial setting?
BINARY: the possible outcomes of each trial can be classified as "success" or "failure" INDEPENDENT: trials must be independent- knowing the outcome of one trial must not have any affect on the result of any other trial NUMBER: the number of trials n of the chance process must be fixed in advance SUCCESS: on each trial, the probability of success must be the same
What is a binomial random variable
The probability distribution of x with parameters n and o, where n=the number of trials of the chance process and p=the probability of success on any one trial. The possible values of x are the whole numbers from 0 to n
When checking for binomial random variables, what should you make sure?
To check BINS and are being asked to count the number of successes in a certain number of trials
How to find binomial probability?
P(x=k)=(number of arrangements of k successes)(p^k)(1-p)^(n-k) OR binomcdf/binompdf on calculator
Formulas for means and standard deviation of a binomial random variable
µ(x)=np Standard deviation(x)= sq.rt. (np(1-p)) ***THESE FORMULAS CAN ONLY BE USED FOR BINOMIAL DISTRIBUTIONS
When can you use a binomial distribution to model the count of successes in the sample if we are sampling without replacement?
When taking an SRS of size n from a population size N, we can use a binomial distribution as long as we don't sample 10% of the population OR the sample size is less than 10% of the total population
When n is large, the distribution of x is approximately normal with mean and standard deviation:
µX=np Standard deviation=sq rt(np(1-p))
We will use the normal approximation when n is so large that np≥10 and n(1-p) ≥10 -> the expected # of successes and failures are both at least 10
When does a geometric setting arise?
When we perform independent trials of the same chance process and record the number of trials until success occurs.
What are the 4 conditions of a geometric setting?
BINARY: the possible outcomes of each trial can be classified as "success" or "failure" INDEPENDENT: trials must be independent- knowing the outcome of one trial must not have any affect on the result of any other trial TRIALS: the goal is to count the number of trials until the 1st success occurs SUCCESS: on each trial, the probability of success must be the same
How to find geometric probability
if y is a geometric random variable with probability p of success on each trial, then how do you find its mean (expected value)?