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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.
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