Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous if its possible values form an interval on a number line probability distribution of a random variable - assignment of probabilities to each possible value of the variable - presented in the form of a table (var. must be discrete) lists values and associated probabilities histogram (var. must be discrete) whose rectangles are given heights that measure the probabilities that values of the variable lie in that class formula represented as a probability distribution function (pdf.) and typically denoted p(x)
Chapter 7 2 Continuous Random Variables probability density curve - continuous analog to a histogram for a discrete random variable - curve must lie above the x-axis - probability that x lies in some interval (written in the form P( a < x < b ) ) is the area under the curve and above that interval - total area under the curve must be 1 - function f(x) whose graph is the density curve is called the probability density function (pdf); in the language of calculus, P ( a < x < b ) = f (x) dx - endpoints of intervals do not affect probability values: P( a < x < b ) = P( a x < b ) = P( a < x b ) = P( a x b ) - the function c(a) = P( x < a ) is the cumulative denisty function (cdf): P( a < x < b ) = c(b) c(a) b a
Chapter 7 3 Statistics associated with a random variable x mean ( µ x ) - locates the center of the distribution - represents the expected value of x - if x is discrete, - if x is continuous, µ x = x p(x) all x µ x = x f (x) dx variance ( σ x 2 ) - expected squared deviation from the mean - if x is discrete, σ x 2 = - if x is continuous, (x µ x ) 2 p(x) all x σ x 2 = (x µ x ) 2 f (x) dx standard deviation ( σ x ) - measure of the typical deviation from the mean - square root of the variance
Chapter 7 4 Linear combinations of random variables If x is a random variable, y = a + bx defines another random variable which is a linear function of x mean of y is variance of y is µ y = µ a+bx = a +b µ x σ 2 2 y = σ a+bx = b 2 σ x 2 standard deviation of y is σ y = σ a+bx = b σ x
Chapter 7 5 More generally, if x 1, x 2,, x n is a collection of random variables and a 1, a 2,, a n is a collection of constants, y = a 1 x 1 + a 2 x 2 + + a n x n defines another random variable, called a linear combination of the x s mean of y is µ y = a 1 µ x1 +a 2 µ x2 + +a n µ xn if the x s are independent, the variance of y is σ 2 y = a 2 1 σ 2 x1 +a 2 2 σ 2 2 x2 + +a n σ xn so, if the x s are independent, the standard deviation of y is σ y = a 2 1 σ 2 x1 +a 2 2 σ 2 2 x2 + +a n σ xn if the x s are dependent, calculation of variance and standard deviation are much more complicated!
Chapter 7 6 The Binomial Distribution binomial experiment - chance experiment consisting of n trials with two possible outcomes, labelled (by convention) success (S) and failure (F) - outcome of each trial is independent of the others - probability of success π is same for each trial binomial random variable - x = number of successes among the n trials of a binomial experiment - probability distribution function is p(x) = P(x successes among the n trials) n! = x!(n x)! π x (1 π) n x n! - The expression x!(n x)! is also denoted n or x nc x and is called a binomial coefficient; it counts the number of ways that x objects can be chosen from a set of n objects [TI83: DIST binompdf( n, π, x ) computes p(x); DIST binompdf( n, π ) computes the entire pdf (all values of p(x), for x = 0,1,,n)]
Chapter 7 7 Statistics for a binomial random variable: mean: µ x = nπ standard deviation: σ x = nπ(1 π) The Geometric Distribution geometric random variable - x = number of trials of a binomial experiment before the first success occurs - p(x) = (1 π) x 1π [TI83: DIST geompdf( π, x ) computes p(x) ]
Chapter 7 8 The Normal Distribution normal random variable - continuous random variable x whose distribution curve is the normal probability curve, a bell-shaped symmetric distribution - where µ and σ are the mean standard deviation of x, the pdf is 1 f (x) = 2 σ 2π e (x µ) 2σ 2 - peak of the normal curve occurs at x = µ, the points of inflection at x = µ ± σ standard normal distribution - normal random variable z with mean µ = 0 and standard deviation σ = 1 - x and z are linearly related by the formulas x = µ + zσ, z = x µ σ - if a and b have z-scores a* and b*, respectively, then P( a < x < b ) = P( a* < z < b* )
Chapter 7 9 [TI83: DIST normalpdf(x, µ, σ ) can be used to graph a normal curve; DIST normalpdf(x) will graph a standard normal curve; similarly, DIST normalcdf( a, b, µ, σ ) computes the normal probability P( a < x < b ) while DIST normalcdf( a*, b* ) computes the standard normal probability P( a* < z < b* ) Note: if a (or a*) is, use the value -1E99 (= 10 99 ) and if b (or b*) is +, use the value 1E99 (= 10 99 ); these are the largest numbers representable in the calculator.]
Chapter 7 10 Extreme values A common situation is one in which a tail probability P( < x < b ) or P( a < x < ) (or P( < z < b* ) or P( a* < z < ) ) for a normal distribution is given and one wishes to know the associated extreme value, either a or b (or a* or b*), of the variable x (or z). [TI83: DIST InvNorm( P, µ, σ ) computes the extreme value b for which P( < x < b ) = P; DIST InvNorm( P ) computes the extreme value b* for which P( < z < b* ) = P ]