_1 x 0.50_ + _2 x 0 by hcj

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```									      Economics 105: Statistics
• Any questions?
• Go over GH 4
Discrete Random Variables
• Take on a limited number of distinct values
• Each outcome has an associated probability
• We can represent the probability distribution function
in 3 ways
– function ƒ(xi) = P(X = xi)
– graph
– table
• Bernoulli distribution
– graph & table ?
• Cumulative distribution function
Discrete Random Variable
Summary Measures
• Expected Value (or mean) of a discrete
distribution (Weighted Average)
N
m X º E(X) = å X i P( X i )
i =1

X     P(X)
– Example: Toss 2 coins,                   0     0.25
X = # of heads,                 1     0.50
compute expected value of X:             2     0.25

E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0
Discrete Random Variable
Summary Measures
(continued)
• Variance of a discrete random variable
N
s X = E[(X - mX ) 2 ] = å[X i - E(X)]2 P(X i )
2

i=1

• Standard Deviation of a discrete random variable

sX = s              2
X

where:
E(X) = Expected value of the discrete random variable X
Xi = the ith outcome of X
P(Xi) = Probability of the ith occurrence of X
Discrete Random Variable
Summary Measures
(continued)

– Example: Toss 2 coins, X = # heads,
compute standard deviation (recall E(X) = 1)

s=        å[X - E(X)] P(X )
i
2
i

s = (0 -1) 2 (0.25) + (1-1) 2 (0.50) + (2 -1) 2 (0.25) = 0.50 = 0.707

= 0, 1, or 2
Properties of Expected Values
• E(a + bX) = a + bE(X), where a and b are constants

• If Y = a + bX, then
var(Y) = var(a + bX) = b2var(X)
Example
• Let C = total cost of building a pool
• Let X = days to finish the project
• C = 25,000 + 900X
• X P(X = xi)
10     .1       Find the mean, std dev, and
11     .3       variance of the total cost.
12     .3
13     .2
14     .1
Permutations and Combinations
• Need to count number of outcomes
• Number of orderings
– x objects must placed in a row
– can only use each once
– x! = (x)(x-1)(x-2) … (2)(1) called “x factorial”
• Permutations
– suppose these x ordered boxes can be filled with n objects
–n>x
– What is the number of possible orderings now?
– Permutations of n objects chosen x at a time = nPx
– nPx = n(n-1)(n-2) … (n-x+1) = n!/(n-x)!
Permutations and Combinations
• How many ways to arrange, in order, 2 letters selected
from A through E?
• What if order doesn’t matter?
• Combinations
nCx   = nPx/x! = n!/ [(n-x)! * x!]
• Eight people (5 men, 3 women) apply for a job. Four
employees are needed. If all combinations are equally
likely to be hired, what is the probability no women will
be hired?
The Binomial Distribution
Probability
Distributions

Discrete
Probability
Distributions

Bernoulli

Binomial

Poisson

Hypergeometric
Binomial Distribution
• Binomial distribution is composed of repeated
Bernoulli trials
•Let X1, X2, …, XN be Bernoulli r.v.’s, then B is
N
B = å Xi
distributed binomially
i =1
•Probability of x successes in N trials is
N -x
P( B = x)= N Cx p (1 - p)
x
, x = 0,1,2,..., N
where p is the prob of “success” on a given trial
Binomial Distribution
• Let B ~ binomial, with p = prob of success, N =
number of trials
• Find E[B] and Var[B] … but first a couple more rules
on the mathematics of expectations with more than 1 r.v.
Two Random Variables
• Expected Value of the sum of two random variables:

E(aX + bY) = aE( X ) + bE(Y )
• Variance of the sum of two random variables:

Var(aX + bY) º s aX+bY = a2s X + b2s Y + 2abs XY
2           2       2

• Standard deviation of the sum of two random
variables:
s aX +bY = s aX +bY
2
Binomial Distribution
• Let B ~ binomial and now find E[B] and Var[B]
• McCoy’s Tree Service in Mocksville, NC removes
dead trees from commercial and residential properties.
They have found that 40% of their invoices are paid
within 10 working days. A random sample of 7 invoices
is checked. What is the probability that fewer than 2 will
be paid within 10 working days?

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