Definition: Probability for Experiments with Discrete Sample Spaces
An experiment has discrete sample space S = cfw_e1, e2, . . The experiment in repeated N times. The observed outcome is ei a fraction fi of the times such that f1 +f2 + . = 1. Definition:
Continuous Random Variables
r.v. X = lifetime of a battery number of observations = 50 number of categories in histogram = 9 interval size 1.0
.3 Freq .2 .1
lifetime in 100-hrs
Function f(x)=.5e-.5x fits the histogram, sort of. From histogram, add block
Expected Value of a Continuous Random Variable
The expected value or mean of a continuous random variable with density f(x) is E( X ) = =
xf ( x)dx
x 0.
ex: Find E(X) given f ( x ) = e x E( X ) =
0
x 0 dx + xe x dx =
0
e x 1 1 x x xe e dx = 0 = ( 1) = 0
The Normal Distribution
A continuous normal r.v. X has probability density function f (x ) = 1 e 2
( x) 2
2 2
<x<
parameters: -< and >0. E(X)= and V(X)=2
Normal Distribution
f(x)
x
When Does the Normal Distribution Arise
Measurement is subject to source
Exponential Distribution
Exponential r.v. X with rate : x0 pdf: f ( x ) = e x x0 cdf: F ( x ) = 1 e x moments: E(X)=1/ V(X)=1/2
The Memoryless Property
A component has an exponentially distributed lifetime, X, with mean 10 hours. f ( x ) = .1e .1x x0 P(
Joint Distributions for Discrete R.V.
Consider 2 Discrete r.v. 's r.v. X with possible values x = x1, x2, . r.v. Y with possible values y = y1, y2, . The joint distribution function is a list of outcomes of X and Y: (x1, y1), (x1, y2), (x1, y3), (x2, y1)
Mean and Variance of a Linear Function of n independent random variables
Given: Xi, i = 1.n, independent random variables E(Xi), V(Xi) i=1.n Linear function Y = c 0 + c 1 X 1 + . . . .+ c n X n The mean and variance of Y: E( Y ) = c 0 + c 1 E( X 1 )+ . .
Sample Averages
Xi, i = 1.n independent identically distributed (iid) random variables from a population with mean and standard deviation Sample Average is a random variable! r.v.
X= X1 + . . . . + X n n
The mean and variance of the average 1 1 E( X ) =
Point and Interval Estimates
Goal: Estimate the parameter of a distribution ex: Estimate from a Normal distribution ex: Estimate p from a Binomial distribution Steps: 1. Collect a random sample of size n 2. Compute estimate of the parameter 3. Identify th
Use the Law of Total Probability to Solve Problems
A company orders boxes of 10 machined parts from two suppliers. Supplier 1 supplies 70 percent and has defective rate .03. Supplier 2 supplies 30 percent and has defective rate .05. Find the probability
Three Commonly Used Probability Distributions for Discrete Random Variables
Dist Binomial
Random Var. r.v. X = number of successes in n independent trials r.v. X = number of failures until first success r.v. X = number of events in the interval
Possible V
FUNCTION OF A DISCRETE R.V.
r.v. X= number of defects on a part. Here is prob dist of X: x 012 f(x) .3 .5 .2
In each example, r.v. C = cost Give function c=h(x) and prob dist of C.
Ex) The cost per unit is $2 per defect. Ex) The cost is $3 to check a unit
Conditional Probability
example: S = all Rutgers students E = engineers W = women consider these different probabilities P(W) = .5 P(E) = .05 P(W / E) = .2 P (E / W ) = .02 P(E W) = .01
Definition of Conditional Probability
given 2 events E S and F S P( E
Total Probability Rule
Weighted Average: There are two suppliers A and B. A supplies 60% of parts awith fraction defective .02. B supplies 40% of parts with fraction defective .05. Find the fraction of defective parts.
2 Suppliers with Notation
There are
Bayes' Theorem
A supplies 60% of parts with fraction defective 0.02. B supplies 40% of parts with fraction defective 0.05. events event D = part is defective event A = part comes from supplier A event B = part comes from supplier B Given P(A) =.6 P(B) =.4
A Discrete Random Variable, X, is a function that assigns a number to each outcome in sample space, S
ex: flip a coin twice
S = cfw_ HH, TH . HT, TT
r.v. X = number of heads Outcome r.v. X HH 2 TH 1 HT 1 TT 0 Possible values for r.v. X are 0, 1, and 2
You
Expected Value of a Discrete Random Variable
discrete r.v. X possible values x1, x2, . pdf f(x)=P(X=x)
E ( X ) = xi f ( xi ) = xi P ( X = xi )
i i
Called: Expected Value of X Mean of X E(X)
What is difference between "average" and mean?
Mean = Populati
Common Discrete Probability Distribution Functions The Binomial r.v.
Experiment: There are N independent trials. The outcome for each trial is success with probability p and failure with probability 1-p. r.v. X = number of successes in N independent trial
Geometric Distribution
experiment: Conduct independent trials. The outcome on each trial is success with probability p and failure with probability 1-p. Stop at the first success. S = cfw_ s, fs, ffs, fffs, ffffs, fffffs, . . . random variable: r.v. X =
Poisson Distribution
r.v. X = number of events in an interval Probability Distribution Function:
e x f ( x) = P ( X = x) = x!
0,1, .
Parameter: = E(no. of events in interval) Mean and Variance: E( X ) = V(X ) =
Examples
r.v. X = number of defects (not
Probability - The Basics
Repetitive operation experiment Outcome in unknown in advance Set of all possible outcomes is known
Sample Space is SET of all Possible Outcomes, S
ex) experiment: flip a coin
S=cfw_h, t
ex) experiment: machine fills a 12 ounce ca