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33 Cards in this Set
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random variable
(df) 
a random variable is a variable that assumes NUMERICAL VALUES associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample


example of random variable

Discover a frequency distribution for the number of students that use the walkup ATM on campus each day over the course of a year


What is the experimental unit

the ATM


What is the experiment

counting the # of students using the ATM of a particular day


What is the random variable

# of students


What are the sample points for this random variable?

0,1,...# of students at UNR


Discrete random variables
(df) 
Discrete random variables can assume a countable number of values
i.e. has to be a whole integer, doesn't use fractions. ex. you can't cast half of a vote, can't have 3.5 visits to the ATM 

Continuous random variables
(df) 
Continuous random variables can have values corresponding to any of the points contained in one or more intervals
i.e. can be in between integers, can go to fractions 

Examples of discrete random variables

1. whether a hunter gets an elk tag in 2007
2. the number of semesters a student takes to graduate in your major 3. whether a driver gets caught speeding 

Examples of continuous random variables

1. a swimmer's time in the 50 meter butterfly
2. the time it takes a student to do the apst270 homework 3. the amount of acreage burned in wildfires within a 100 mile radius of Reno each summer 

When studying a discrete random variable, we want to know:
(2) 
1. the possible values it can have
2. the probability of each value 

Values can be expressed as
(2) 
fractions or percentages
ex. .25, 25% 

The probability distribution of a discrete random variable
(df) 
The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume


Example of finding a probability distribution

1.Experiment: tossing 2 coins simultaneously
2. random variable x: # of heads observed 3. x can assume values of 0, 1, and 2 HH: x=2, prob=.25 TH: x=1, prob=.25 HT: x=1, prob=.25 TT; x=0, prob=.25 

Requirements for the probability distribution of a discrete random variable x
(2) 
1. p(x) must be greater than or equal to 0 for all values of x
2. The sum of the probabilities must equal 1 (no probability is greater than 1, and probability can't be negative) 

signs of x and p(x)

x can be neg,
p(x) can't be negative 

If you don't know N, calculate the mean using the formula:

mean equals the sums of p(x) * x (for all values of x)
note: mean can be neg 

Formula for the mean, or expected value of a discrete random variable

u (mu) = E(x) = sum of x * p(x)


Formula for the variance of a discrete random variable

o^2 = sum of (xu)^2 * p(x)


Formula for standard deviation

o = square root of o^2


When we know the mean (u) and standard deviation (o) of the probability distribution of x,

1. we can use Chebyshev's rule
2. and the Empirical rule to make statements about the likelihood of values of x falling within the intervals: (u +/ o), (u +/ 2o), (u +/ 3o) 

Why do we use those intervals?

We use those intervals because we often want to describe HOW FAR an observation is away from its mean in terms of the standard deviation


Example of using the intervals

Suppose your score on the midterm is 70%. You hear that the mean was 50% and the standard deviation was 10%. Your score is 2 standard deviations ABOVE the mean


Chebyshev's Rule

applies to any data set, regardless of the shape of the frequency distribution. The rule states that the minimum amount of data that will lie within K (k>1) standard deviation of the mean for any distribution of data.
There will be at least 75% of the data within 2 standard deviations of the mean and at least 88% of the data within 3 standard deviations of the mean. 

Empirical Rule

only works for bellshaped symmetric data
1. aboout 68% of the values lie within one standard deviation of the mean 2. about 95% of the values will lie within 2 standard deviations of the mean 3. about 99.7% of the values will lie within 3 standard deviations of the mean 

Binomial Random Variable
(characteristics) 
1. An experiment of n identical trials
2. 2 possible outcomes on each trial, denoted as S(success) and F(failure) 3. Probability of success (p) is constant from trial to trial. Probability of failure (q) is 1p or P^c 4. Trials are independent 5. Binomial random variable  number of S's in n trials 

Multiplicative Rule using example numbers

P(3 of the 4 adults pass the test) = 4(.1)^3(.9)


Formula for the probability distribution p(x)

p(x) = (n over x) * p^x * q ^(nx)
where p = probability of success on single trial q=1p n= number of trials x= number of successes in n trials 

Example where n = 4, x = 0

p(0) = (4 over 0) *p^0 * q^4


Formula for the mean

u = np


formula for the Variance

o^2 = npq


Formula for standard deviation

o = square root of (n*p*g)


Instructions for making a binomial distribution on Minitab

calc
>probability > binomial a. select probability, enter # of trials, and probability of success, input where to put calculations (i.e. c1) then do bar chart to show data 