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cps720: Probability Formula Sheet

1 Conditional Probability

is the probability of the occurence of event given that occured and is given as

Knowing that occured reduces the sample space to , and the part of it where also occured

is . Note that this formula is well defined only if ! # .

Because is commutative, we have

When random events ) are mutually exclusive and exhaustive set of events (ie, events ) and 1 cannot co-occur if

3 4

6 and8

)

) is the whole sample space), then

A

)

) A

)

) )

If random events E and F are independent, we have and thus

That is, knowledge that whether F has occured does not change the probability that E occurs.

2 Probability Distribution and Density Function The probability distribution function P of a random variable R for any real number S is

S

T

R V S X

and we have T

S ` R V c X

c

f

S

If R is a discreet random variable then

S

A

h i q r

t

where P is the function defined as S

T

R

S X, and T

R

S Xmeans the probability

of the event that a random variable R has the value S in a random experiment.

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3 Conditional Expectation

Expectation, or expected value, or mean of a discrete random variable R , denoted by R , is

the sum of the values that R can take multiplied by the probabilities that each value can occur

R

A

)

t ) t )

where t ) is one of the possible values of R and R ) U T R t ) X is the probability of

the event that the variable R takes the valuet )

in a random experiment. In other words, it is aweighted average where each value is weighted by the probability that R takes that value.

It is a well-known result in probability theory that, if S and c are real numbers, and R and

are random variables, then

S R c

S

R c (1)

R

R

(2)

The conditional expectation of a discrete random variable R given that a ramdom variable

takes the value

, i.e.

, is defined by

R

A

)

t ) T

R

t )

X

Define a random variable R that takes values R 1 with probability T 1 X .

Because the conditional expectation R is a random variable, if is a discrete random

variable, then by the definition of expectation we can compute the expected value of

R

as

R

A

1

R

1

T

1

X

P

Its also a standard result that

R

R

A

1

R

1

T

1

X

This equation states that to calculate R , we can take a weighted average of the conditional

expected value of R given that 1 , each of the terms R 1 being weighted by the

probability T

1

Xof the event on which it is conditioned.

4 Weak Law of Large Numbers

Let R T R ! X #! % &

be a set of ( independent and identically distributed (iid) random variables each

having mean ) and a finite variance 0 1 . Then for any 3 ! #

4 5

5

5

5 7

!

R !

(

f

)

5

5

5

5

!

3 @ B

#

S D ( B F

That is, the average value of R over ( trials converges to the mean value of R as ( increases.

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5 Special Distributions

5.1 Uniform Distribution

A random variable R is uniformly distributed over the interval [a, b] if its density function is

given by

t

4

c

f

S

ifS V

t

V c

# otherwise

If R is uniform, its expected value and variance are

R

S c

S

R

c

f

S

1

5.2 Normal (Gaussian) Distribution

A random variable R is normally distributed with mean ) and variance 0 1 , denoted as ( )

01

,

if its density function is

t

0

t

f

t f

)

1

0

1

f

F `

t

` F

6 Generating Random Numbers in C

The following C library functions can be used to generate a random value according to a uni-

form distribution between # and

:

#include

int rand(void);

void srand(unsigned int seed);

Do a man rand on the command line to see more details about these functions

To generate a random number according to the Gausian distribution, the GNU Scientific Library

can be used. For information about how to use this library, about Unix random generators, or

about the Gaussian random functions, check out these links

http://www.gnu.org/software/gsl/manual/gsl-ref_2.html

http://www.gnu.org/software/gsl/manual/gsl-ref_17.html#SEC272

http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC288

7 Generating Random Numbers in JavaInformation related to random number generating in Java can be found at

http://java.sun.com/j2se/1.5.0/docs/api/java/util/Random.html

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