• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Probability amplitude wikipedia, lookup

Law of large numbers wikipedia, lookup

History of statistics wikipedia, lookup

Inductive probability wikipedia, lookup

Foundations of statistics wikipedia, lookup

Transcript
```Discrete Probability Distributions
6- 1
Chapter
Six
McGraw-Hill/Irwin
Random variable
A result from an experiment that, by chance, can
take different values.
Egs.
1. No. of heads you would get in 3 tosses of a coin
2. No. of employees absent on a shift
3. No. of students at CSUN in a semester
4. No. of minutes to drive home from CSUN
5. Inches of rainfall in LA during a year
6. Tire pressure in PSI of a car tire
Examples 1-3 are ‘discrete’
4-6 are ‘continuous’ random variables
In this chapter, we focus on the ‘discrete’.
Probability
Distribution
A listing of all possible outcomes of an
experiment and the corresponding probability.
c
P
F
T
Possible Outcomes
N
= 1/8
= 3/8
= 3/8
= 1/8
Probability Distribution
Note the similarity to histogram
Mean of Discrete Probability Distribution
  [ xP( x)]
where
  represents the mean
o x is each outcome
o P(x) is the probability of the various outcomes x.
Calculating Mean/Expected Value of Heads in 3
coin-toss experiment
  [ xP( x )]
μ = 0 * .125 + 1 * .375 + 2 * .375 + 3 * .125
= 1.5
(which by intuition makes sense because for each toss you have 50%
 is a weighted average.
It is also referred to as Expected Value, E(X).
Binomial Probability Distribution
•An outcome of an experiment is classified into one of
two mutually exclusive categories, such as a success or
failure (bi means two).
•The data collected are the results of counts (hence, a
discrete probability distribution).
•The probability of success stays the same for each trial
(independence).
Binomial Probability Distribution
P( x)n Cx (1   )
x
n x Can you logically
explain this formula?
n is the number of trials
x is the number of observed successes
π is the probability of success on each trial
Cx 
n
n!
x!(n-x)!
Let us re-visit the 3-toss coin experiment
n=3
π = .5
X = 0,1,2,3 (number of heads)
P(0) = 3C0 * (.5)0 * (1 - .5)3-0
P(1) = 3C1 * (.5)1 * (1 - .5)3-1
P(2) = 3C2 * (.5)2 * (1 - .5)3-2
P(3) = 3C3 * (.5)3 * (1 - .5)3-3
= .125
= .375
= .375
= .125
You can also
use
Appendix A
(page 489)
Practice time!
Do Self-Review 6-3, Page 168
Use Appendix A, Page 490
Cumulative Binomial Probability Distribution
Problem 21, Page 173
Use table in Page 491
a. 0.387 ( n=9; x=9; straight from table )
b. 0.001 ( P(x<5) = P(x ≤ 4) )
c. 0.992 ( 1 – P(x ≤ 5) = 1 – 0.008 )
d. 0.946 [ (P(x=7) + (P(x=8) + (P(x=9) ]
Mean of the Binomial Distribution
  n
Logic: If you throw a coin say 100 times, how many
times you would expect to get a head?
For each throw, the probability of getting a head is .5.
So, in 100 trials, you would expect 50 heads (which is
100*.5; ie. nπ )
Variance of the Binomial Distribution
  n (1   )
2
For the previous example,
= 100*.5*(1-.5)
σ 2 = 25
S.D. σ = 5
```
Related documents