Random variables;Random variables;discrete and continuous discrete and continuous probability distributionsprobability distributions

June 23, 2004June 23, 2004

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)

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

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.

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.

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)

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

Cumulative probabilityCumulative probability

x

P(x)

1/6

1 4 5 62 3

1/31/22/35/61.0

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

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

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

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

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.

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

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

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%

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:

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.

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

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

ExampleExample

2 x

p(x)

1

.865 .135-1 -1 2)P(x 2 e

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

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

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?

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:

Expected Value and Expected Value and VarianceVariance

All probability distributions are characterized by an expected value and a variance (standard deviation squared).

For example, bell-curve (normal) distribution:

Mean One standard deviation from the mean (average distance from the mean)

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.

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

Expected value, formallyExpected value, formally

xall

)( )p(xxXE ii

Discrete case:

Continuous case:

dx)p(xxXE ii xall

)(

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

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

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…

Reading for this weekReading for this week

Walker: 1.1-1.2, pages 1-9