AP Statistics - Chapter 7 Notes Random variable – a variable whose value is a numerical outcome of a random phenomenon (in other words, it is a variable whose value is a randomly determined number) Discrete random variable – a variable with a countable number of possible values Probability Distribution – a list, table or graph showing all of the possible values of a random variable and the probability of each value occurring Probability Histogram – a histogram (recall this is a type of graph) showing the relative frequency (probability) of each possible value a random variable may take on EX: On a test there were 8 As, 6 Bs, 7 Cs and 2 Ds – create a probability histogram for each possible grade – what is the probability a randomly chosen student will have a B? Less than a B? EX: Use your data from the “craps” activity to generate a probability histogram of winning and losing, then generate a probability histogram of each sum – how could we use this to find the probability of winning given a certain roll on the first roll (that does not automatically win or lose * Density curve – a curve that is always on or above the x-axis & has an area underneath it of exactly 1 Continuous random variable – a variable that takes on all possible real number values in a given interval (e.g.; most calculator random number generators will simulate generating a random number which can be any value along the interval (0,1) ) and probability distribution is a density curve, with a continuous random variable, the probability of getting any one specific value is approximately zero, so we talk about the probability of getting a value within a certain range

EX: Unit square – find probability of getting a value between two points

EX: A triangle – find probability of getting a value between two points

EX: A normal distribution is a density curve, so if we take the standardized form Z = (X – µ) / σ of a normally distributed variable X, we have a continuous random variable

Mean of a Random Variable (Discrete) – Think of this as a weighted average. The mean is the sum of each possible value multiplied by the probability that result occurs. Mean of a Random Variable (Continuous) – This is the same thing as finding the mean of a probability distribution (Chapter 2); in other words you should find the “center of gravity” of the distribution, which is the point at which the curve would balance if it were constructed of a physical material Note that even a variable with infinitely many possible values can have a finite expected value, EX: " 1 $2 # n2 = 6 n=1

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Rules for Means: If X is a random variable and a and b are fixed numbers: µa +bX = a + bµX

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If X and Y are random variables, then: µX +Y = µX + µY

! of a Random Variable (Discrete): With a random variable X with a mean of Variance µ, the variance is:

" X2 = # (x i $ µX ) 2 pi

"X =

$ (x

i

# µX ) 2 pi = " X2

Rules for Variances: 1. If X is a random variable and a and b are constants: ! ! 2 " a+bX = b 2" 2X 2. If X and Y are independent random variables: " 2X +Y = " 2X + " Y2 " 2X #Y = " 2X + " Y2 ! This is the addition rule of independent random variables. ! N.B.; Any linear combination of independent Normal random variables is also Normally ! distributed.

Law of Large Numbers – Assume you will draw independent observations at random from a large population with mean µ. Decide how accurately you want the estimate of µ to be. As the number of observations increases, the mean x of the sampled values eventually approaches the mean µ of the whole population as closely as you specified and then stays that close. N.B.; The law of large numbers does not say how many trials are needed to get “close ! enough” to the population mean, just that you can get close enough if you do enough trials. The Myth of the “Law of Small Numbers”: See the “hot hand” basketball example from section 7.2 in the book.