Chapter 5: Probability Distributions: Discrete Probability Distributions
Random variables; discrete and continuous probability distributions June 23, 2004.
-
date post
22-Dec-2015 -
Category
Documents
-
view
228 -
download
3
Transcript of Random variables; discrete and continuous probability distributions June 23, 2004.
![Page 1: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/1.jpg)
Random variables;Random variables;discrete and continuous discrete and continuous probability distributionsprobability distributions
June 23, 2004June 23, 2004
![Page 2: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/2.jpg)
Random VariableRandom Variable
• A random variable x takes on a defined set of values with different probabilities.
• For example, if you roll a die, the outcome is random (not fixed) and there are 6 possible outcomes, each of which occur with probability one-sixth.
• For example, if you poll people about their voting preferences, the percentage of the sample that responds “Yes on Kerry” is a also a random variable (the percentage will be slightly differently every time you poll).
• Roughly, probability is how frequently we expect different outcomes to occur if we repeat the experiment over and over (“frequentist” view)
![Page 3: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/3.jpg)
Random variables can be Random variables can be discrete or continuousdiscrete or continuous
Discrete random variables have a countable number of outcomes Examples:• Binary: Dead/alive, treatment/placebo, disease/no disease,
heads/tails• Nominal: Blood type (O, A, B, AB), marital
status(separated/widowed/divorced/married/single/common-law)
• Ordinal: (ordered) staging in breast cancer as I, II, III, or IV, Birth order—1st, 2nd, 3rd, etc., Letter grades (A, B, C, D, F)
• Counts: the integers from 1 to 6, the number of heads in 20 coin tosses
![Page 4: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/4.jpg)
Continuous variableContinuous variable
A continuous random variable has an infinite continuum of possible values. – Examples: blood pressure, weight, the speed of a car,
the real numbers from 1 to 6. – Time-to-Event: In clinical studies, this is usually how
long a person “survives” before they die from a particular disease or before a person without a particular disease develops disease.
![Page 5: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/5.jpg)
Probability functionsProbability functions A probability function maps the possible values of
x against their respective probabilities of occurrence, p(x)
p(x) is a number from 0 to 1.0. The area under a probability function is always 1.
![Page 6: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/6.jpg)
x
p(x)
1/6
1 4 5 62 3
Discrete example: roll of a dieDiscrete example: roll of a die
xall
1 P(x)
![Page 7: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/7.jpg)
Probability mass functionProbability mass function
x p(x)
1 p(x=1)=1/6
2 p(x=2)=1/6
3 p(x=3)=1/6
4 p(x=4)=1/6
5 p(x=5)=1/6
6 p(x=6)=1/61.0
![Page 8: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/8.jpg)
Cumulative probabilityCumulative probability
x
P(x)
1/6
1 4 5 62 3
1/31/22/35/61.0
![Page 9: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/9.jpg)
Cumulative distribution Cumulative distribution functionfunction
x P(x≤A)
1 P(x≤1)=1/6
2 P(x≤2)=2/6
3 P(x≤3)=3/6
4 P(x≤4)=4/6
5 P(x≤5)=5/6
6 P(x≤6)=6/6
![Page 10: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/10.jpg)
ExamplesExamples
1. What’s the probability that you roll a 3 or less? P(x≤3)=1/2
2. What’s the probability that you roll a 5 or higher? P(x≥5) = 1 – P(x≤4) = 1-2/3 = 1/3
![Page 11: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/11.jpg)
In-Class ExercisesIn-Class ExercisesWhich of the following are probability functions?
1. f(x)=.25 for x=9,10,11,12
2. f(x)= (3-x)/2 for x=1,2,3,4
3. f(x)= (x2+x+1)/25 for x=0,1,2,3
![Page 12: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/12.jpg)
In-Class ExerciseIn-Class Exercise
1. f(x)=.25 for x=9,10,11,12
Yes, probability function!
x f(x)
9 .25
10 .25
11 .25
12 .25
1.0
![Page 13: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/13.jpg)
In-Class ExerciseIn-Class Exercise
2. f(x)= (3-x)/2 for x=1,2,3,4
x f(x)
1 (3-1)/2=1.0
2 (3-2)/2=.5
3 (3-3)/2=0
4 (3-4)/2=-.5
Though this sums to 1, you can’t have a negative probability; therefore, it’s not a probability function.
![Page 14: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/14.jpg)
In-Class ExerciseIn-Class Exercise
3. f(x)= (x2+x+1)/25 for x=0,1,2,3
x f(x)
0 1/25
1 3/25
2 7/25
3 13/25
Doesn’t sum to 1. Thus, it’s not a probability function.
24/25
![Page 15: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/15.jpg)
In-Class Exercise:In-Class Exercise: The number of ships to arrive at a
harbor on any given day is a random variable represented by x. The probability distribution for x is:
x 10 11 12 13 14P(x)
.4 .2 .2 .1 .1
Find the probability that on a given day:
a. exactly 14 ships arrive
b. At least 12 ships arrive
c. At most 11 ships arrive
p(x=14)= .1
p(x12)= (.2 + .1 +.1) = .4
p(x≤11)= (.4 +.2) = .6
![Page 16: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/16.jpg)
In-Class Exercise:In-Class Exercise:You are lecturing to a group of 1000 students. You ask them to each randomly pick an integer between 1 and 10. Assuming, their picks are truly random:
• What’s your best guess for how many students picked the number 9?
Since p(x=9) = 1/10, we’d expect about 1/10th of the 1000 students to pick 9. 100 students.
• What percentage of the students would you expect picked a number less than or equal to 6?
Since p(x≤ 5) = 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 =.6 60%
![Page 17: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/17.jpg)
Continuous caseContinuous case The probability function that
accompanies a continuous random variable is a continuous mathematical function that integrates to 1. For example, recall the negative
exponential function (in probability, this is called an “exponential distribution”):
xexf )(
1100
0
xx ee
This function integrates to 1:
![Page 18: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/18.jpg)
Continuous caseContinuous case
x
p(x)
1
The probability that x is any exact particular value (such as 1.9976) is 0; we can only assign probabilities to possible ranges of x.
![Page 19: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/19.jpg)
For example, the probability of x falling within 1 to 2:
23.368.135. 2)xP(1 122
1
2
1
eeee xx
x
p(x)
1
1 2
![Page 20: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/20.jpg)
Cumulative distribution Cumulative distribution functionfunction
As in the discrete case, we can specify the “cumulative distribution function” (CDF):
The CDF here = P(x≤A)=
AAAA
xA
x eeeeee 110
00
![Page 21: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/21.jpg)
ExampleExample
2 x
p(x)
1
.865 .135-1 -1 2)P(x 2 e
![Page 22: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/22.jpg)
Example 2: Uniform Example 2: Uniform distributiondistribution
The uniform distribution: all values are equally likely
The uniform distribution:f(x)= 1 , for 1 x 0
x
p(x)
1
1
We can see it’s a probability distribution because it integrates to 1 (the area under the curve is 1): 1011
1
0
1
0
x
![Page 23: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/23.jpg)
Example: Uniform distributionExample: Uniform distribution
What’s the probability that x is between ¼ and ½?
x
p(x)
1
1¼ ½
P(½ x ¼ )= ¼
![Page 24: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/24.jpg)
In-Class ExerciseIn-Class ExerciseSuppose that survival drops off rapidly in the year following diagnosis of a certain type of advanced cancer. Suppose that the length of survival (or time-to-death) is a random variable that approximately follows an exponential distribution with parameter 2 (makes it a steeper drop off):
]1102:[
2)( :functiony probabilit
0
2
0
2
2
xx
T
eenote
eTxp
What’s the probability that a person who is diagnosed with this illness survives a year?
![Page 25: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/25.jpg)
AnswerAnswerThe probability of dying within 1 year can be calculated using the cumulative distribution function:
The chance of surviving past 1 year is: P(x≥1) = 1 – P(x≤1)
)(2
0
2 1)( TT
x eeTxP
135.)1(1 )1(2 e
Cumulative distribution function is:
![Page 26: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/26.jpg)
Expected Value and Expected Value and VarianceVariance
All probability distributions are characterized by an expected value and a variance (standard deviation squared).
![Page 27: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/27.jpg)
For example, bell-curve (normal) distribution:
Mean One standard deviation from the mean (average distance from the mean)
![Page 28: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/28.jpg)
Expected value of a random Expected value of a random variable variable
If we understand the underlying probability function of a certain phenomenon, then we can make informed decisions based on how we expect x to behave on-average over the long-run…(so called “frequentist” theory of probability).
Expected value is just the weighted average or mean (µ) of random variable x. Imagine placing the masses p(x) at the points X on a beam; the balance point of the beam is the expected value of x.
![Page 29: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/29.jpg)
Example: expected valueExample: expected value
Recall the following probability distribution of ship arrivals:
x 10 11 12 13 14P(x)
.4 .2 .2 .1 .1
5
1
3.11)1(.14)1(.13)2(.12)2(.11)4(.10)(i
i xpx
![Page 30: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/30.jpg)
Expected value, formallyExpected value, formally
xall
)( )p(xxXE ii
Discrete case:
Continuous case:
dx)p(xxXE ii xall
)(
![Page 31: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/31.jpg)
Extension to continuous Extension to continuous case:case:
example, uniform random example, uniform random variablevariable
x
p(x)
1
1
2
10
2
1
2)1()(
1
0
21
0
x
dxxXE
![Page 32: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/32.jpg)
In-Class ExerciseIn-Class Exercise
3. If x is a random integer between 1 and 10, what’s the expected value of x?
5.5)1(.552
)110(10)1(.
10
1)
10
1()(
1010
1
ii
iixE
![Page 33: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/33.jpg)
Variance of a random variableVariance of a random variable
If you know the underlying probability distribution, another useful concept is variance. How much does the value of x vary from its mean on average?
More on this next time…
![Page 34: Random variables; discrete and continuous probability distributions June 23, 2004.](https://reader035.fdocuments.us/reader035/viewer/2022062320/56649d775503460f94a5923d/html5/thumbnails/34.jpg)
Reading for this weekReading for this week
Walker: 1.1-1.2, pages 1-9