Statistical Techniques B – Notes

1

Discrete Random Variables and Probability Distributions 

!  A random variable is a variable that takes on numerical values realized by the outcomes in the

sample space generated by a random experiment.

!  A random variable is discrete if it can take on no more than a countable number of values.

!  A random variable is continuous if it can take any value in an interval.

!  The probability distribution function, P (x ), of a discrete random variable X   represents the

probability that X  takes the value x , as a function of x .

!  The probability distribution function of a discrete random variable must satisfy the following two

properties:

1.  0!P (x )! 1 for any value of x .  

2.  The individual probabilities sum to 1.

!  The cumulative probability distribution of a random variable X  represents the probability that

X  does not exceed a value, i.e. F (x 0)=P (X  ! x 

0).  

!  The properties of cumulative probability distributions for discrete random variables are quite similar

to that of the normal one.

!  The expected value of a discrete random variable X  is defined as:

-  E [X ]= µ = xP (x ).x 

!  

!  The variance of a discrete random variable is defined as:

-  ! 2= E [(X !µ)2 ]= E [X 

2 ]!µ2= (x !µ)2P (x ).

x 

"  

!  Summary of properties for linear functions of a random variable:  

- µY = E [a +bX ]= a +bµ

X 

! Y 

2=Var (a +bX )= b2

! X 

2 

!  Summary results for the mean and variance of special linear functions:  

-  If a random variable always takes the value a , it will have mean a  and variance 0.

!  The number of sequences with x  successes in n  independent trials is:

-  C x 

n =

n !

x !(n ! x )! 

!  If n   independent trials are carried out, the distribution of the number of resulting successes, x , is

called the binomial distribution: 

- P (x )=n !

x !(n ! x )!P 

x (1!P )n !x  for x = 0,1,2,…,n .  

!  Let X  be the number of successes in n  independent trials, each with probability of success P . Then X  

follows a binomial distribution with mean and variance follows:

- µ = E [X ]= nP 

! X 

2= E [(X !µ

X )2 ]= nP (1!P )

 

!  Assumptions of the Poisson distribution: 

 

Statistical Techniques B – Notes

2

-  The probability of the occurrence is constant for all subintervals.

-  There can be no more than one occurrence in each subinterval.

-  Occurrences are independent; that is, an occurrence is one interval does not influence the proba-

bility of an occurrence in another interval.

!  A random variable X  is said to follow the Poisson distribution if it has the probability distribution:

-  P (x )=e !!!x 

x !, for x = 0,1,2,… 

-  The constant !  is the expected number of successes per time or space unit. And it is the mean 

and variance of the distribution.

!  The Poisson distribution can be used to approximate the binomial probabilities when n  is large and

P  is small (preferably such that ! = nP ! 7 ):

-  P (x )=e !nP (nP )x 

x ! for x = 0,1,2,…  

!  It can also be shown that when n ! 20 and P " 0.05 , and the population mean is the same, both the

binomial and the Poisson distributions generate approximately the same probability values.

!  Suppose that random sample of n  objects is chosen from a group of N  objects, S  of which are suc-

ceses. The distribution of X  (number of successes) is the hypergeometric distribution: 

-  P (x )=C x 

s C 

n !x 

N !s /C n 

N  

!  Let X  and Y  be a pair of discrete random variables. Their joint probability distribution express-

es the probability that simultaneously X  takes the value x  and Y  takes the value y :

-  P (x ,y )= P (X = x !Y = y ).  

!  In jointly distributed random variables, the probability distribution of the random variable X  is its

marginal probability distribution , which is obtained by summing the joint probabilities over all

possible values: P (x )= P (x ,y ).y 

!  

!  The conditional probability distribution of Y  given that X  takes the value x  is defined as:

-  P (y | x )= P (x ,y ) /P (x ).  

!  The jointly distributed random variables X  and Y  are independent if: P (x , y ) = P (x )P (y ).

!  The conditional mean is computed using the following: µY |X = E [Y |X ]= (y | x )P (y | x ).

y 

!  

!  The expectation of any function g (X , Y ) of these random variables is defined as follows:

-  E [g (X ,Y )]= g (x ,y )P (x ,y ).y 

!x 

!  

!  The expected value of (X !µX )(Y  ! µ

Y ) is called the covariance between X  and Y .

- Cov (X ,Y )= E [(X !µ

X )(Y  !µ

Y )]=   (x !µ

X )(y !µ

Y )P (x ,y )

y 

"x 

"   .

Cov (X ,Y )= E [XY ]!µX µY .

 

!  The correlation between X  and Y  is as follows: ! =Corr (X ,Y )=Cov (X ,Y )/ " X " Y .  

!  If two random variables are statistically independent, the covariance between them is 0. Howev-

er, the converse is not necessarily true.