Economics 105 statistics
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Economics 105: Statistics. Any questions? Go over GH 3 & 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(x i ) = P(X = x i ) graph table

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Economics 105: Statistics

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Economics 105 statistics

Economics 105: Statistics

Any questions?

Go over GH 3 & 4


Discrete random variables

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

Discrete Random Variable Summary Measures

  • Expected Value (or mean) of a discrete

    distribution (Weighted Average)

    • Example: Toss 2 coins,

      X= # of heads,

      compute expected value of X:

      E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0

X P(X)

0 0.25

1 0.50

2 0.25


Discrete random variable summary measures1

Discrete Random Variable Summary Measures

(continued)

  • Variance of a discrete random variable

  • Standard Deviation of a discrete random variable

    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 measures2

Discrete Random Variable Summary Measures

(continued)

  • Example: Toss 2 coins, X = # heads,

    compute standard deviation (recall E(X) = 1)

Possible number of heads = 0, 1, or 2


Properties of expected values

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

Example

Let C = total cost of building a pool

Let X = days to finish the project

C = 25,000 + 900X

XP(X = xi)

10 .1Find the mean, std dev, and

11 .3 variance of the total cost.

12 .3

13 .2

14 .1


Permutations and combinations

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 combinations1

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?


Economics 105 statistics

The Binomial Distribution

Probability Distributions

Discrete

Probability Distributions

Bernoulli

Binomial

Poisson

Hypergeometric


Binomial distribution

Binomial Distribution

Binomial distribution is composed of repeated Bernoulli trials

Let X1, X2, …, XN be Bernoulli r.v.’s, then B is distributed binomially

Probability of x successes in N trials is

where p is the prob of “success” on a given trial


Binomial distribution1

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

Two Random Variables

  • Expected Value of the sum of two random variables:

  • Variance of the sum of two random variables:

  • Standard deviation of the sum of two random variables:


Binomial distribution2

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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