Random Variables. Intro to discrete random variables. Random Variables. “A random variable is a numerical valued function defined over a sample space” What does this mean in English? If Y rv then Y takes on more than 1 numerical value Sample space is set of possible values of Y
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Random Variables
Intro to discrete random variables
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Random Variables
Deterministic variables
Continuous
random variables
Discrete
random variables
Variables are
but models
Variables
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P = S  C
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Note
conventions!
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Expected value of a function of y, a discrete rv
Let g(y) be function of y
Suppose C = g(T) = 5T + 3, find E(C)
E(C) = [5(4)+3]0.2 + [5(5)+3]0.3 + [5(6)+3]0.5 = 29.5
Let d = constant
E(d)= constant
E(dy)= dE(y)
E() is a linear operator
E(X + Y) = E(X) + E(Y), where X & Y are rv
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Variance of a discrete rv
Previously defined variance for population & sample
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How can we derive these?
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We know that
and
We also know that
So it follows that
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Problem:
Toss coin 3 times, find P(2 heads)
n = 3 ; y = 2
P(H, H, T) = (.5)(.5)(.5) = 0.125
Could also be (H,T,H) or (T, H, H)
P(2 heads) = 0.125 + 0.125 + 0.125 = 0.375
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Returning to the P(2 heads)
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y n
Probability of y successes
# of combinations
Probability of n  y failures
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Have 20 coin tosses
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Problem
Events E1, …, Ek occur with probabilities p1, p2, …, pk . Given n independent trials probability E1 occurs y1 times, … Ek occurs yk times.
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Need to understand convention
Note there are k random variables
This is called
a joint distribution.
j = npj j2 = npjqj = npj(1pj)
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æ
ö
y
1

ç
÷
=
r
y
r
p
(
y
)
p
q
ç
÷

r
1
è
ø
r
rq
m
=
=
s 2
2
p
p
Let p = 0.5 & r = 2, do we get
reasonable results?
Problem
Have series of Bernoulli trials, want probability of waiting
until yth trial to get rth success
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Hypergeometric Probability Distribution
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1) Why is y a rv?
2) What do we mean by
p(y)?
3) What is r/N ?
N # in population
n # in sample
r # of Successes in population
y # of Successes in sample
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# of defects in an 8x8 sheet of plywood
# of cars passing a fixed point in one minute
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Why does this
make sense?
Note particularly
interesting relationship
Note must be for
the same unit of
measure!
Where,
y # of occurrences in a given unit
mean # of occurrences in a given unit
e 2.71828…
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Special
Functions
HYPGEOMDIST
BINOMDIST
NEGBINOMDIST
POISSON
Are there others?
Excel
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