AP Statistics Ch 7: Random Variables

Before starting, you should know...

• sample spaces don’t have to be made of numbers, but in statistics, we are more interested in numbers, so we assign a variable X = the number a result is happening
• X varies as we do trials
• we call X a random variable, or a variable whose value is a numerical outcome of a random phenomenon
• usually represent random variable with capital letter near end of alphabet
• we are most interested in the mean of the random sample
• sample spaces of random variables are just possible number of times random variable can occur; we do not mention sample space separately
• we also mention probability that each event of X will occur
• will learn two types of probability models and two ways to assign probabilities to values of X

Part 1: Discrete and Continuous Random Variable

• discrete random variables
• countable number of values
• values can be any rational value; the important thing to remember is that it isn’t every possible value between number A and number B
• probability still follow same rules: any probability must be between 0 to 1, and the sum of the probabilities of an distribution must be 1
• you can find probabilities of an event by adding probabilities of components that make up that event
• probability histogram
• like any other histogram, but for discrete variables, it is made of blocks; DO NOT make the curve of histogram smooth for discrete variables
• outcomes on x-axis, probability on y-axis
• area under histogram = 1
• great for comparing distributions
• Continuous Random Samples
• infinite number of values
• contains any possible value between two numbers
• to find probability of section in an event, we can’t add individual probabilities of components together to get result, b/c too much values; basically, an individual outcome (a single value) will be assigned a probability of 0
• so then P(X>a) and P(x is greater or equal to a) are equivalent
• ONLY TRUE FOR CONTINUOUS RANDOM VARIABLES!!!
• modeled by a density curve (area of 1), so we will measure probability using area under section of density curve instead
• Using normal distributions as probability distributions
• normal distributions and density curves can be used to describe probabilities, so they can be considered as probability curves
• we can use z-scores to calculate probability

Part 2: Calculating means and variances of random variables

• discrete random value
• mean
• symbolized by Greek letter mux
• often not possible value of X
• also called expected value
• is what we expect in the long run
• how to find it: http://www.youtube.com/watch?v=QmILCjisiZM
• is a weighted average; this means it is weighted by its probability; this makes sure that we take into account the fact that some outcomes are not as likely to occur; average of possible values
• variance
• symbolized by o--x2
• how to find it:  http://www.youtube.com/watch?v=2vXmLLKqGig
• for standard deviation, just square it
• estimation
• to estimate mean, choose SRS, then use mean from SRS to estimate mean of event; this makes the estimated mean a random variable b/c it can vary with every SRS
• law of large numbers makes it safe to use SRS mean to estimate mean of random sample
• as more samples are drawn, the mean of the SRS will get closer and closer to the mean of the random variable
• so mean of SRS sample gets close to the mean of event and the proportion of the number of times something will happen gets closer to the probability that the “something” will happen; this rule describes what will happen in the long run
• true for any distribution
• more about law of large numbers
• outcomes that occur over short time (few trials) won’t have same behavior as that when outcomes that occur in events that takes place over a long time (more trials)
• events w/ a lot of trials are more accurate
• nothing needs to happen for an event to keep its probability; if a chain of similar outcomes happen that seem to throw off the probability, other outcomes don’t need to occur to shift probability back into place because its probability never shifted in the first place; probability only accurate in the long run
• the more variated the outcomes of the event are, the more trials are needed to make the mean of SRS accurate
• rules for means
• to find total mean of two means, just add them together
• if the question asks you to find a mean in another unit, perform a transgression using a model that converts the current unit to the desired unit; y = mx + b, where x is the current mean with the current unit and y is the wanted mean
• rules for variances
• association between variances makes us unable to add them together to find the total standard deviation
• for independent events, add the variances together to get the total variance, then square root to get standard deviation
• works for both the sum of variances and the difference of variances; use the same equation to calculate for total or difference in variations or standard deviations
• total (or difference of) standard deviation 2 = (standard deviation 1)2 + (standard deviation 2)2
• remember, this is only for INDEPENDENT EVENTS!
• for change of units, only the thing that is multiplied to units affects standard deviation and variance
• use the equation o-- = b2(standard deviation)2
• b is what you are multiplying by
• remember, variances can be added together, but standard deviations can’t
• combining normal random variables
• if we combined or subtracted independent normal variations linearly, then the new distribution is also normally distributed
• still use mean and standard deviation rules (see above) to calculate the mean and standard deviation of new distribution
• combined/subtracted linearly = both random variables are multiplied by a constant
• aX + dY, or aX - dY, where a and d are constants and X and Y are random variables

awesome website for Statistics that I found: https://www.khanacademy.org/math/probability/random-variables-topic/random_variables_prob_dist/v/discrete-and-continuous-random-variables

list of videos for calculating mean and variance of random variables (mentioned in notes above)

discrete variables (variance): http://www.youtube.com/watch?v=2vXmLLKqGig

discrete variables (mean): http://www.youtube.com/watch?v=QmILCjisiZM

continuous variables (mean and variance): http://www.youtube.com/watch?v=gPAxuMKZ-w8