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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(xi) = P(X = xi)

graph

table

Bernoulli distribution

graph & table ?

Cumulative distribution function

- 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

- Example: Toss 2 coins,

X P(X)

0 0.25

1 0.50

2 0.25

(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

(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

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

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

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 distributed binomially

Probability of x successes in N trials is

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.

- 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 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?