Transcript Correlation - People Server at UNCW

• you are familiar with the term “average”, as in arithmetical average of a
set of numbers (test scores for example) – we used the symbol
X , or " X  bar " to stand for this average. In probability, we have an
“average” value too – the so-called mathematical expectation, which is
defined intuitively as the “long-run” average value of the possible values
of an experiment. In particular, if the possible values of a probability
experiment are a1, ... , ak and the corresponding probabilities of these
values occurring are
p1, ... , pk then the expectation (E) is the weighted average of the values,
with the weights being the probabilities:
E  a1 p1  a2 p2  ... ak pk
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Illustrate this with the “relative frequency” approach to probabilities...
Example: “Pick 3” lottery – payoff is $500, cost of ticket is $1
HW: page 95, #3.81-3.86
Review exercises (p.97-99): #3.98, 3.100-3.104, 3.107
• We are going to be concerned with random variables, a
function that assigns a numerical value to each possible
outcome of a random experiment
• X=sum of spots when a pair of fair dice is thrown
• Y=# of Hs that come up when a fair coin is tossed 4 times.
• We are going to be interested in the probability distribution of
the random variable (rv): i.e., the list of all the possible
values of the rv and the corresponding probabilities that the
variable takes on those values... get the probability
distributions of the two example rvs X and Y above...
• If we write f(x) = P(X = x) then it must be the case that
f ( x)  0 and
 f ( x)  1
all x
• We will be considering several discrete random variables and their
distributions
• Binomial: X = the number of S’s in n Bernoulli trials
(recall Bernoulli trials or see p. 105)... our previous rv counting the
number of Hs in 4 tosses of a fair coin is a typical Binomial variable
– we denote the binomial probabilities by
n x
b( x; n, p)    p (1  p) n  x , for x  0,1,..., n
 x
– Cumulative binomial probabilities are given in Table 1 for various values
x
of n and p; write
B( x; n, p)   b(k ; n, p)
k 0
– TI-83 can do binomial probabilities under 2nd VARS binompdf (individual
binomial probs) or binomcdf (cumulative binomial probs)
– R can calculate binomial probabilities (see the R handout)
• we can think about the binomial rv as “sampling with
replacement” (Hs and Ts in equal numbers in an urn,
replace after each draw...). The hypergeometric rv can
be thought about in the same way except that the
sampling is to be done “without replacement” from an
urn with N items, a Ss and N-a Fs.
 a  N  a 
 

x
n

x
 for x  0,1,..., n
h( x; n, a, N )   
N
 
n
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Lot of 20 items, 5 defective (“success”). Choose 10 at random; what is the
probability that exactly 2 of the 10 are defective?
Ans: h(2; 10, 5, 20) = .348
NOTE: This can be approximated by b(2; 10, .25) = .282 (not so good...but if
the N were larger... what would happen?)
• Example (p.111) N=100, a=25 (so p remains .25):
then h(2; 10, 25, 100) = .292 and b(2; 10, .25) = .282
• The mathematical result is that as N approaches infinity
with p=a/N, h(x; n, a, N) approaches b(x; n, p) and a
good rule of thumb is to use the binomial if n <= N/10
• HW: Read sections 4.1-4.3; work on problems # 4.2,
4.3, 4.5, 4.7, 4.9, 4.13, 4.17, 4.19, 4.20, 4.21, 4.23, 4.25,
4.28, 4.29