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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)
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 curve is called
the probability density function (pdf); in
the language of calculus,
P( a < x <b ) = Úab 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)
Statistics associated with a random variable x
• mean ( mx )
- locates the center of the distribution
- represents the expected value of x
- if x is discrete,
m x = Â x ⋅ p(x)
all x
- if x is continuous,
†
•
m x = Ú-•
x ⋅ f (x) dx
• variance ( s x2 )
- expected
† squared deviation from the mean
- if x is discrete,
†
s x2 = Â (x - m x )2 ⋅ p(x)
all x
- if x is continuous,
†
•
s x2 = Ú-•
(x - m x )2 ⋅ f (x) dx
• standard deviation ( sx )
- measure of the typical deviation from the mean
†
- square root of the variance
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
m y = m a+bx = a + b ⋅ m x
• variance of y is
†
2
s 2y = s a+bx
= b 2 ⋅ s 2x
• standard deviation of y is
†
†
s y = s a+bx = |b | ⋅s x
More generally, if x1, x2, … , xn is a collection of random
variables and a1, a2, … , an is a collection of constants,
y = a1x1 + a2x2 + … + anxn defines another random
variable, called a linear combination of the x’s
• mean of y is
m y = a1m x1 + a 2m x 2 +L+ an m xn
• if the x’s are independent, the variance of y is
†
s 2y = a12s x21 + a22s x22 +L+ an s x2n
• if the x’s are independent, the standard deviation of
y is
†
s y = a12s x21 + a 22s x22 +L+ a ns x2n
• if the x’s are dependent, calculation of variance and
standard deviation are much more complicated
†
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 p 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!
=
p x (1- p )n-x
x!(n - x)!
Ên ˆ
n!
- The expression
is also denoted Á ˜ or
x!(n
x)!
Ëx ¯
†
nCx 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,º,n )]
†
Statistics for a binomial random variable:
• mean:
mx = np
• standard deviation:
sx = np (1- p )
†
The Hypergeometric Distribution
†
• hypergeometric random variable
- x = number of successes chosen in a sample of n
choices selected without replacement from a
population of N objects (choices are not
independent)
- when n is no more than 5% of N, the
hypergeometric distribution is very closely
approximated by a binomial disribution
The Geometric Distribution
• geometric random variable
- x = number of trials of a binomial experiment
before the first success occurs
- p(x) = (1 – p)x–1p
[TI83: DIST geompdf( p , x ) computes p(x) ]
The Normal Distribution
• normal random variable
- continuous random variable x whose
distribution curve is the normal probability
curve, a bell-shaped symmetric distribution
- where m and s are the mean standard deviation
of x, the pdf is
1
-( x-m )2 2s 2
f (x) =
e
s 2p
- peak of the normal curve occurs at x = m, the
points of inflection at x = m ± s
†
• standard normal distribution
- normal random variable z with mean m = 0 and
standard deviation s = 1
- x and z are linearly related by the formulas
x-m
x = m + zs , z =
s
- if a and b have z-scores a* and b*, respectively,
then P( a < x < b ) = P( a* < z < b* )
†
[TI83:
DIST normalpdf(X, m , s ) can be used to graph a
normal curve; DIST normalpdf(X) will graph a
standard normal curve;
similarly, DIST normalcdf( a , b , m , s ) 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 (= –1099)
and if b (or b*) is +•, use the value 1E99 (= 1099); these
are the largest numbers representable on the
calculator.]
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
the extreme value, either a or b (or a* or b*), of the
variable x (or z) associated with this given probability is
desired
[TI83: DIST InvNorm( P , m , s ) 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 ]