Random variables; discrete and continuous probability distributions June 23, 2004

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Random variablesdiscrete and continuous
probability distributionsJune 23, 2004
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Random 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)

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Random variables can be discrete 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/c
  ommon-law)
• Ordinal (ordered) staging in breast cancer as I,
  II, III, or IV, Birth order1st, 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

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Continuous 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.

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Probability 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.

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Discrete example roll of a die
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Probability mass function
1.0
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Cumulative probability
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Cumulative distribution function
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Examples
1. Whats the probability that you roll a 3 or
less? P(x3)1/2 2. Whats the probability
that you roll a 5 or higher? P(x5) 1
P(x4) 1-2/3 1/3
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In-Class Exercises

• Which of the following are probability
  functions?
• 1.      f(x).25 for x9,10,11,12
• 2.      f(x) (3-x)/2 for x1,2,3,4
• 3. f(x) (x2x1)/25 for x0,1,2,3

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In-Class Exercise

• 1.      f(x).25 for x9,10,11,12

x f(x)
9 .25
10 .25
11 .25
12 .25
1.0
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In-Class Exercise

• 2.      f(x) (3-x)/2 for x1,2,3,4

x f(x)
1 (3-1)/21.0
2 (3-2)/2.5
3 (3-3)/20
4 (3-4)/2-.5
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In-Class Exercise

• 3. f(x) (x2x1)/25 for x0,1,2,3

x f(x)
0 1/25
1 3/25
2 7/25
3 13/25
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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

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(x14) .1
p(x?12) (.2 .1 .1) .4
p(x11) (.4 .2) .6
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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
• Whats your best guess for how many students
  picked the number 9?
• Since p(x9) 1/10, wed 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

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Continuous 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)

• This function integrates to 1

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Continuous case
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.
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For example, the probability of x falling within
1 to 2
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Cumulative distribution function
As in the discrete case, we can specify the
cumulative distribution function (CDF) The
CDF here P(xA)
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Example
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Example 2 Uniform distribution
The uniform distribution all values are equally
likely The uniform distribution f(x) 1 , for
1? x ?0
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Example Uniform distribution
 Whats the probability that x is between ¼ and
½?
P(½ ?x? ¼ ) ¼
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In-Class Exercise
Suppose 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)
Whats the probability that a person who is
diagnosed with this illness survives a year?
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Answer
The probability of dying within 1 year can be
calculated using the cumulative distribution
function  
  Cumulative distribution function is
The chance of surviving past 1 year is P(x1)
1 P(x1)
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Expected Value and Variance

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

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For example, bell-curve (normal) distribution
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Expected value of a random 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.

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Example expected value

• Recall the following probability distribution of
  ship arrivals

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Expected value, formally
Discrete case
Continuous case
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Extension to continuous caseexample, uniform
random variable
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In-Class Exercise
3. If x is a random integer between 1 and 10,
whats the expected value of x?
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Variance 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

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Reading for this week

• Walker 1.1-1.2, pages 1-9