Further distributions

Discrete random variables

1

We know that for a discrete random variable .

The function that allocates probabilities, , is known as

the , sometimes abbreviated .

iall x

P X x p

P X x

X Xprobability density function of p.d.f. of

( ) ( )

(expected value or mean)i iall x

E X x P X x x p

2 2( ) ( ) ( )

e.g. all x

E g X g x P X x E X x P X x 2 2 2( ) ( ) ( ) ( ) (variance of ); (standard deviation)Var X E X E X X Var X

2

2

( ) ( ) 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

E a a Var a

E aX aE X Var aX a Var X

E aX b aE X b Var aX b a Var X

,

( ) ( )

The , sometimes abbreviated

is given by .F x P X x X Xcumulative distribution function of c.d.f. of

Expectation and variance

2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

In general, for random variables and and constants and

If and are , then

X Y a b

E aX bY aE X bE Y

E aX bY aE X bE Y

X Y

Var aX bY a Var X b Var Y

independent

2 2( ) ( ) ( ) Var aX bY a Var X b Var Y

1 2

1 2

1 2

, ,

( ) ( )

( ) ( )

Taking observations from we have

n

n

n

n X X X X

E X X X nE X

Var X X X nVar X

independent

Read Examples 4.18-4.20, pp.257-260 Do Q1-Q7, p.261

Linear combinations of normal variables

2 21 1 2 2

2 21 2 1 2

2 21 2 1 2

2

( , ) ( , )

,

,

( ,

For two independent normal variables such that and

For independent normal variables such that i i i

X N Y N

X Y N

X Y N

n X N

2 2 21 2 1 2 1 2

2

21 2

2

)

,

( , )

,

( ,

For independent observations of the random variable where

For the normal variable such that

n n n

n

X X X N

n X X N

X X X N n n

X N

2 2

2 21 1 2 2

2 2 2 21 2 1 1 1 2

)

( , )

( , ) ( , )

( , )

, and for any constant

For two independent normal variables such that and

and for any constants and

a

aX N a a

X N Y N

a b

aX bY N a b a b

2 2 2 2

1 2 1 1 1 2( , ) aX bY N a b a b

Read Examples 8.1-8.11, pp.403-417 Do Q1-Q4, p.417

The Poisson distributionConsider the following random variables:

the number of emergency calls received by an ambulance centre in an hour,

the number of vehicles approaching a freeway toll bridge in a five-minute interval,

the number of flaws in a metre length of silk,

the number of cancer cells on a slide.

Assuming that each occurs randomly, these are all examples of variables that

can be modelled using a

Poisson dist .ribution

If

events occur singly and at random in a given interval of time or space,

, the mean number of occurences in the given interval, is known and is finite,

the variable is the number of oX

( ) ( ) 0,1,2,3,!

ccurences in the given interval,

then

Po , where for to infinity.xe

X P X x xx

Properties of the Poisson distribution

( ) ( ) 0,1,2,3,!

Po , where for xe

X P X x xx

( ) ( 0) ( 1)Po and X P X e P X e

( ) ( ) ( )Po and X E X Var X

1

If a Poisson distribution has an integer mean, then the distribution

is bimodal, with modes and .

If a Poisson distribution does not have an integer mean, then the

mode is the largest integer below

.

Read Examples 5.18-5.21, pp.292-295 Do Q1-Q10, pp.297-298

The sum of independent Poisson variables

( ) ( )

( )

For independent variables, and , if Po and Po ,

then Po

X Y X m Y n

X Y m n

Read Examples 5.25 & 5.26, pp.301-302 Do Exercise 5f, p.303

Continuous random variablesA continous random variable is given by its p.d.f. with a

particular domain.

X

This function may be represented by a (non-negative) curve.

Probabilities may be calculated as an area under the curve

using integration or other geometric properties.

1

21 2

( )

( ) 1

( )

For a continuous random variable , with p.d.f. , valid

over the domain ,

(a)

(b) for ,

b

a

x

x

X f x

a x b

f x dx

a x b P x X x f x dx

Read Examples 6.1 – 6.4, pp.315-319

Do Exercise 6a, pp.319-320

Expectation

2 2

( )

( ) ( )

( ( )) ( ) ( )

( ) ( )

For a continuous random variable with p.d.f. ,

In particular,

all x

all x

all x

f x

E X x f x dx

E g X g x f x dx

E X x f x dx

( )

( ) ( )

( ) ( )

E a a

E aX aE X

E aX b aE X b

Read Examples 6.5 – 6.7, pp.320-323

Do Exercise 6b, pp.323-324

Read Examples 6.8 – 6.10, pp.325-327

Variance and mode

2 2

( )

( ) ( )

( ) ( )

( )

For a continuous random variable with p.d.f. ,

,

where

The standard deviation of is often denoted by i.e. .

all x

all x

f x

Var X x f x dx

E x x f x dx

X Var X

2

2

( ) 0

( ) ( )

( ) ( )

Var a

Var aX a Var X

Var aX b a Var X

( )The mode is the value of for which is greatest in the given

domain for .

X f x

XRead Examples 6.11 – 6.15, pp.328-333

Do Exercise 6c, pp.333-334

Cumulative distribution function

( ) ( )

For a particular value, , in the domain of the function

.t

t

F t P X t f x dx

( )

( ) ( )

lower limit

So if is valid in the range ,

then

t

a

f x a x b

F t f x dx

( )

( )

1

Note that

b

a

F b P X b

f x dx

1 2 2 1( ) ( )P x X x F x F x

( )If is a continuous random variable with p.d.f , the

can be found by integrating.

X f x cumulative

distribution function (c.d.f.) F(x)

Median, quartiles and other percentiles

2 ( )

( ) 0.5 ( ) 0.5

If is the median, , of the distribution, then for defined in ,

i.e. m

a

m Q f x a x b

f x dx F m

1

1

1

( )

( ) 0.25 ( ) 0.25

If is the lower quartile of the distribution, then for defined in ,

i.e. Q

a

Q f x a x b

f x dx F Q

3

3

3

( )

( ) 0.75 ( ) 0.75

If is the upper quartile of the distribution, then for defined in ,

i.e. Q

a

Q f x a x b

f x dx F Q

100

In general, th n

F n percentile

Read Examples 6.16 & 6.17, pp.336-339

Do Exercise 6d, pp.339-341

Cumulative distribution function

( ) ( )

For a particular value, , in the domain of the function

.t

t

F t P X t f x dx

( )

( ) ( )

lower limit

So if is valid in the range ,

then

t

a

f x a x b

F t f x dx

( )

( )

1

Note that

b

a

F b P X b

f x dx

1 2 2 1( ) ( )P x X x F x F x

( )If is a continuous random variable with p.d.f , the

can be found by integrating.

X f x cumulative

distribution function (c.d.f.) F(x)

( ) ( ) :

( ) ( )

To obtain from

.

f x F x

df x F x F x

dx

Do Exercise 6e, pp.343-344

Read Examples 6.18 - 6.20, pp.341-343

Uniform distribution

1( )

, .

The probability density function for a continuous random variable,

distributed uniformly on the domain is

.

This is denoted by

a x b

f xb aX R a b

2

,

2

( ).

12

If , show that

and

X R a b

a bE X

b aVar X

Do Exercise 6f, pp.349-350

Read Examples 6.21 - 6.26, pp.345-349

,

0

1

If , then

X R a b

x a

x aF x a x b

b ax b

The p.d.f. of a related variableFrequently, the random variable being measured is not the ultimate objective. What many be of primary interest is some function of these variables.

If is some function of :Y X

p.d.f. of X

c.d.f. of X

c.d.f. of Y

p.d.f. of Y

Example 1

6 (1 ) 0 1

0

2 1

If has p.d.f.

elsewhere

and , what is the p.d.f. of ?

X

x x xf x

Y X Y

1 3

1 1

0 1 2 1

10

2

6 (1

The nontrivial case occurs when .

y

F y P Y y P Y P Y y

P X y

yP X

x x

( 1) 2

0

12 3 2

0

3 2

)

3 2

3 91

4 2 4

y

y

dx

x x

y y y

2

0 1

3 93 1 3

4 40 3

So

y

yf y y y

y

Example 2

2

11 2

30

If has p.d.f.

elsewhere

and , what is the p.d.f. of ?

X

xf x

Y X Y

0

0 1 1 4

Here, the set of 's that get mapped into the interval has a different

form depending on whether or .

x Y y

y y

0 1y 21

3 3

y

y

yF y P Y y P y X y dx

1 4y 1

111

3 3

y yF y P Y y P X y dx

0 0

20 1

3

11 4

31 4

y

yy

F yy

y

y

10 1

3

11 4

6

0 elsewhere

yy

f y yy

Questions( ) 3 4 Let have the uniform p.d.f. over the unit interval. Find , where .X f y Y X Q1:

0.1

ln

Suppose that has p.d.f. ,

Find the p.d.f. for .

Find the p.d.f. for .

xX f x e x

YX

Y X

Q3 :

(a)

(b)

6 1 0 1 ( ) 2 3 If , , find , where .f x x x x f y Y X Q2 :

2 23 0 1 ( ) 4 If has p.d.f. , , find , where .X f x x x f y Y X Q4 :

The geometric distributionFor a situation to be described using a

independent trials are carried out

the outcome of each trial is deemed either a success or a failure

the probability, , of a successful op

geometric model

utcome is the same for each trial

The discrete random variable, , is the

. If the above conditions are satisfied, is said

to follow a , denoted Geo

X

X

X

number of trials needed to obtain

the first successful outcome

geometric distribution .p

1

1 :

( )

1,2,3,

Writing failure as , where

If Geo , the probability that the first success is obtained at the

th attempt is where

, r

p q q p

X p

r P X r

P X r q p r

0

1

Note: cannot take the value

the number of trials could be infinite

the mode of this distribution is

X

Read Examples 5.2, pp.273 - 274

The geometric distribution

2

1 :

1

If Geo and

,

X p q p

qE X Var X

p p

1 :

1

If Geo and

x

x

X p q p

P X x q

P X x q

Read Examples 5.3 -5.6, pp.274 - 276

Do Exercise 5a, pp.276 - 277

Deriving the (negative) exponential distribution

0

Consider a sequence of independent events occuring at random

points in time at a rate ; i.e. a Poisson process with parameter .

We examine this process at an arbitrary time and denote the

random va

t

riable "the time to the first event" by .X

(0, )no events occur in the time interval P X x P x

0

(

Note that the mean number of events occuring in a time interval of length is ,

and that the probability of obtaining the value from a Poisson distribution

with mean is:

x x

x

0)

0!

xxx e

e

1 1

So:

and:

x

x

P X x e

F X P X x e

Differentiating yields: 0

0 otherwise

xe xf x

The exponential distributionThe distribution of the time intervals between Poisson events is known as

the . This distribution has p.d.f.:exponential distribution

, 0xf x e x

0

0

0

1

For , the c.d.f. is given by:

b x

bx

b

b

F b e dx

e

e

and so:

1

( )

x

x

a b

P X x e

P X x e

P a X b e e

Example 1

Shape of the exponential distribution

1

0

Just as the geometric distribution has a mode of ,

the exponential distribution has a mode of .

1

An unusual property of this distribution is that a lump of it can be thrown away

and the remainder (when rescaled to an area of ) is the same as the original.

| i.e.

This is known as the property of a Poisson process. The probability

of the next event not occurring in the next units of time is independent of

P X a b X a P X b

b

lack of memory

everything

that has occurred up till now (time ).a

( )

|

a b

a

b

P X a b X aP X a b X a

P X a

P X a b

P X a

e

e

e

P X b

Expectation and variance of the exponential distribution

2

1

1

E X

Var X

Example 1

Example 2

Questions1.

2.

3.

4.

5.

6.