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
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 ?
Cumulative distribution function
distribution (Weighted Average)
X= # of heads,
compute expected value of X:
E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0
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
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)
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.
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”
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?
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
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
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.
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?