variance and mean of a distribution powerpoint presentation

7/25/2019 Variance and mean of a distribution powerpoint presentation

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MEAN ANDVARIANCE OF ADISTRIBUTION


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Discrete Uniform Distribution

Discrete Uniform Distribution isa probability distribution whereby afinite number of equally spaced values

are equally likely to be observed;every one of nvalues has equalprobability 1/n.

A simple example of the discreteuniform distribution is throwing a

fair die.


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Continuous Uniform Distribution

Continuous Uniform Distribution

orRectangular Distribution is a family of

probability distributions such that for each

member of the family, all intervals of the same

length on the distribution's support are equallyprobable.


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Example -UniformDistribtion

The uniform distribution: all values are equally likely.

f(x)=1, for 1x 0

x

p(x)

1

1

We can see its a probability distribution because it

integrates to 1 the area under the curve is 1!:

1011

1

0

1

0=== x


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Distribtion

Whats the probability thatxis bet"een 0 and #$

%# x0!& #

x

p(x)

1

1#0


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Expected Value and Variance

All probability distributions arecharacterized by an expected value

(mean) and a variance (standarddeviation squared).

Expected value is an extremely

useful concept for good decision-maing!


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Example! T"e #otter$

A certain lottery "ors by picing #numbers from $ to %&. 't costs $.

to play the lottery* and if you "in*you "in + million.

If you play the lottery once, what are

your expected winnings or losses?


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Lottery

'(

)*

+

10-.

'1+,*'/,1/

1

+)/)*

11===

x$ p(x)

-$ .&&&&&&&+,

+ million 7.2 x 10--8

alculate the probability of "inning in 1 try:

The probability function note, sums to 1.0!:

2)* choose +3

4ut of )* numbers,this is the number of

distinct combinations

of +.


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Expe%te& Vale

x$ p(x)

-$ .&&&&&&&+,

+ million 7.2 x 10--8

The probability function

5pected 6alue

57! & %"in!89,000,000 %lose!8(91.00& .0x 1068 7.2 x 10-8+ .999999928 (-1) =.144 - .999999928

= -$.86;egative epected value is never good


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Expe%te& Vale =f you play the lottery every "eek for 10 years,

"hat are your epected "innings or losses$


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Expe%te& Vale=f you play the lottery every "eek for 10 years,

"hat are your epected "innings or losses$

>0 (.'+! & (9))-.0


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Expe%te& Vale of a

Ran&om VariableExpected value is ust the average ormean (/) of random variablex.

't0s sometimes called a 1"eightedaverage2 because more frequent valuesof 3 are "eighted more highly in the

average.'t0s also ho" "e expect 3 to behave on-

average over the long run (1frequentist2

vie" again).


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Expe%te& Vale

dx)p(xxXE ii= all

!

= ,all! )p(xxXEii

?iscrete ase:

ontinuous ase:


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S$mbol Interl&e

E(3) 4 /5hese symbols are used

interchangeably


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Example6 Expected 7alue

8onsider the follo"ing 9robability:istribution6

x $ $$ $+ $; $%(x) .% .+ .+ .$ .$

=

=++++=>

1

/.11!1.1)!1.1/!.1!.11!).10!i

i xpx


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Example - Expe%tation

hat is the

Expected 7alue of gain if a personpurchases one ticet ?


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Example - Expe%tation

hat is the Expected

7alue of "inning a prize if a personpurchases t"o ticets ?


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Varian%e'Stan&ar& De(iation

+ 4 7ar(x) 4 E(x-)+

2

5he expected (or average) squareddistance (or deviation) from themean3

=== ,all

!@!A! )p(xxxExVar ii


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Varian%e

= ,all

!! )p(xxXVar ii

?iscrete ase:

ontinuous ase:

dx)p(xxXVar ii = all

!!


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S$mbol Interl&e7ar(3)4 +

@:(3) 4 these symbols are usedinterchangeably


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)ra%ti%e )roblem

't costs $. to play the

lottery. 5he probability of "inningthe lottery is $,;, and losing the

lottery is +;,. 5he mean (already

calculated) is -.=;. >hat0s the

variance of! ?


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)ra%ti%e )roblem

@tandard deviation is .&&. 'nterpretation6 /.1!/'B1'!0>/.1

!/'B0!0>/.1!/'B1'!0>/.1

=

+=

++=++=

**.**-. ==


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T"eorem

5he variance of a random variable3 is

C+ 4 E(3+) - /+


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Example

Het the random variable 3 represent thenumber of defective parts for a machine

"hen ; parts are sampled from a productionline and tested. 8alculate 7ariance of thefollo"ing 9robability :istribution of 3.

x * + , f(x) .=$ .;, .$ .$


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Problem 1

Find the mean and the variance ofthe random variable with probability

function or density f!x"#f!x" $ %x, !& x ("


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Problem 3


function f!x"# $ )umber a fair die turns up


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Problem 5


function or density f!x"#*niform distribution on + &, -


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Uniform Distribtion

f(x) 4 $b-a a I x I b


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Problem 7

hat is the expected daily profit, if

a store sells air conditioners per daywith probability f!(&" $ &.(, f!((" $ &./,

f!(%" $ &.0, f!(/" $ &.%, and profit perair conditioner is 122 3


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Problem 9

4f the mileage !in multiples of (&&&miles" after which a tyre must be replaced

is given by the random variable withdensity f!x" $ 5 e65x !x 7 &". hat mileage

can you expect to get on one of thesetyres3 Also find the probability that a tyre

will last at least 0&,&&& miles.


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Problem 11

A small filling station is supplied withgasoline every 8aturday afternoon.

Assume that its volume of sales in ten

thousands of gallons has the probability

density f!x" $ 9x ! ( : x ", if & x ( and &

otherwise. etermine the mean, variance

and the standardi


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Stan&ar&i.e& Ran&om Variable

K 4 3 - /

C

@td Dand 7ariable 4 3 - Bean

@td :eviation


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Problem 13

=et +cm- be the diameter of bolts in aproduction. Assume that has the density

f!x" $ k ! x : &.> " ! (.( : x " if &.> ? x ? (.(

and & otherwise. etermine k, sketch f!x",

and find @ and %.


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ASSI/NMENT NO +LrieMy describe follo"ing "ith the help of examples6Dole of 9robability

Delation bet"een 9robability and @tatistical'nference

Dandom @ampling

Layes0 Dule

Noint 9robability :istribution

Oand >ritten assignments be submitted by $= Pov+$+.

8opying and Hate @ubmissions "ill have appropriatepenalty.

variance and mean of a distribution powerpoint presentation

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