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Random variables
Petter Mostad
2005.09.19
Repetition
• Sample space, set theory, events, probability
• Conditional probability, Bayes theorem,
independence, odds
• Random variables, pdf, cdf, expected value,
variance
Some combinatorics
• How many ways can you make ordered
selections of s objects from n objects?
Answer: n*(n-1)*(n-2)*…*(n-s+1))
• How many ways can you order n objects?
Answer: n*(n-1)*…*2*1 = n! (”n faculty”)
• How many ways can you make unordered
selections of s objects from n objects?
Answer: n  (n  1)   (n  s  1)
n
n!
s!

 
s !(n  s)!  s 
The Binomial distribution
• Bernoulli distribution: One experiment, with
probability π.
• If you have n independent experiments, each with
probability π, what is the probability of s
successes?
• To compute this probability, we find the
probability of getting successes in exactly a
particular set of experiments, and multiply with
the number of choices of such sets.
The Binomial distribution
The random variable X has a Binomial
distribution if it has the distribution as
above:
n s
ns
P( X  s)     (1   )
s
E ( X ) n
Var ( X ) n (1   )
Example
• In a clinic, 20% of the patients come
because of problem X. If the clinic treats 20
patients one day, what is the probability that
zero, one, or two patients will come because
of problem X?
The Hypergeometric distribution
• Assume N objects are given, and s of these
are ”successes”. Assume n objecs are
chosen at random. The distribution of the
number of successes among these is the
hypergeometric distribution:
 s  N  s 
 

x
n

x

P( x)   
N
 
n
Example
• A class consists of 20 girls and 10 boys. A
group of 5 students is selected at random.
What is the probability that it will contain 0,
1, or 2 girls?
The Poisson distribution
• Assume ”successes” happen independently,
at a rate λ per time unit. The probability of
x successes during a time unit is given by
the Poisson distribution:
 x
e 
P( x) 
x!
E( X )  
Var ( X )  
Example
• Assume patients come to an emergency
room at an average rate of 1 per hour. What
is the probability that 3 or more patients
will come within a particular hour?
The Poisson and the Binomial
• It is possible to construct a Binomial variable quite
similar to a Poisson variable:
– Subdivide time unit into n subintervals.
– Set the probability of success in each subinterval to λ/n.
– Letting n goto infinity, the two processes become
identical.
• The above argument can be used to find the
formula for the Poisson distribution.
• Also: Poisson approximates Binomial when n is
large and p is small.
Bivariate distributions
• Probability models where the outcomes are
pairs (or vectors) of numbers are called
bivariate (or multivariate) random variables:
–
–
–
–
P ( x, y )  P ( X  x  Y  y )
marginal probability: P( x)   P( x, y )
P ( x, y )
P( x | y ) 
conditional probability:
P( y )
X and Y are independent if for all x and y:
y
P ( x, y )  P ( x ) P ( y )
Example
• The probabilities for
– A: Rain tomorrow
– B: Wind tomorrow
are given in the following table:
No wind
Some wind Strong wind
Storm
No rain
0.1
0.2
0.05
0.01
Light rain
0.05
0.1
0.15
0.04
Heavy rain
0.05
0.1
0.1
0.05
Covariance and correlation
• Covariance measures how two variables
vary together:
Cov( X , Y )  E ( X  E( X ))(Y  E(Y ))  E( XY )  E( X ) E(Y )
• Correlation is always between -1 and 1:
Corr ( X , Y ) 
Cov( X , Y )
 XY
Cov( X , Y )

Var ( X )Var (Y )
Properties of the expectation and
variance
•
•
•
•
•
•
E ( X  Y )  E ( X )  E (Y )
E (aX  b)  aE ( X )  b
Var (aX )  a 2Var ( X )
If X,Y independent, then E ( XY )  E ( X ) E (Y )
If X,Y independent, then Cov( X , Y )  0
If Cov(X,Y)=0 then
Var ( X  Y )  Var ( X )  Var (Y )
Continuous random variables
• Used when the outcomes are best modelled
as real numbers
• Probabilities are assigned to intervals of
numbers; individual numbers generally
have probability zero
Cdf for continuous random variables
• As before, the cumulative distribution
function F(x) is equal to the probability of
all outcomes less than or equal to x.
• Thus we get P(a  X  b)  F (b)  F (a)
• The probability density function is however
b
now defined so that
P(a  X  b)   f ( x)dx
• We get that
F ( x0 ) 
a
x0


f ( x)dx
Expectations
• The expectation of a continuous random
variable X is defined as
E ( X )   xf ( x)dx
• The variance, standard deviation,
covariance, and correlation are defined
exactly as before, in terms of the
expectation, and thus have the same
properties
Example: The uniform distribution
on the interval [0,1]
• f(x)=1
• F(x)=x 1
1
1
1 2
• E ( X )   xf ( x)dx   xdx   2 x   12
0
0
0
2
2
Var
(
X
)

E
(
X
)

E
(
X
)
•
1
  x d ( x)   0.5  13  14  121
2
0
2
The normal distribution
• The most used continuous probability
distribution:
– Many observations tend to approximately
follow this distribution
– It is easy and nice to do computations with
– BUT: Using it can result in wrong conclusions
when it is not appropriate
The normal distribution
• The probability density function is
f ( x) 
•
•
•
•
1
2 2
e
 ( x   )2 / 2 2
where E ( X )   Var ( X )   2
Notation N (  ,  2 )
Standard normal distribution N (0,1)
Using the normal density is often OK unless
the actual distribution is very skewed
Normal probability plots
Normal Q-Q Plot of Household income in thousands
4
3
2
Expected Normal
• Plotting the quantiles
of the data versus the
quantiles of the
distribution.
• If the data is
approximately
normally distributed,
the plot will
approximately show a
straight line
1
0
-1
-2
-3
-200
0
200
400
600
Observed Value
800
1 000
1 200
The Normal versus the Binomial
distribution
• When n is large and π is not too close to 0 or 1,
then the Binomial distribution becomes very
similar to the Normal distribution with the same
expectation and variance.
• This is a phenomenon that happens for all
distributions that can be seen as a sum of
independent observations.
• It can be used to make approximative
computations for the Binomial distribution.
The Exponential distribution
• The exponential distribution is a distribution for
positive numbers (parameter λ):
f (t )  e t
• It can be used to model the time until an event,
when events arrive randomly at a constant rate
E (T )  1/ 
Var (T )  1/  2