Mathematics Upper Secondary4

8/13/2019 Mathematics Upper Secondary4

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Mathematics upper-secondary4

Distributions and Integration

Discrete random variablesXSIQ

* Intermediate Mathematics - Random variables

Random variables

A random variable [1] is a variable whose value is a numerical outcome of some

random experiment.

[] An experiment is random if the outcome is uncertain.

!or example" an experiment could be the random selection of a male from a

#roup of students $probabilit% &ei#ht 1'()" or the selection of a mussel from a

catch made in order to measure the lead content" or the toss of a coin" or the

rollin# of a die in a board #ame or the selection of a card from a dec of cards.

!or example" with a #roup of male students three different bo%s mi#ht have

hei#hts of 1+, cm" 1' cm and 1'cm.

A random variable has an unnown value before the experiment and an

observed numerical value after the experiment.

Random variables are denoted b% a capital letter" such as X"Yand Z./he observed values are denoted b% the correspondin# lower case letters" x" %"

and 0.

!or example" when a die is rolled" let the random variable" X" be the number on

the face which is uppermost when the die comes to rest.

/here are onl% possible values for X2 1" " 3" " , or . X cannot tae values

lie 1. or 3.'.

/husX is a discrete random variable.

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7ins2 ------ [1] http288alison.com89 [] http://alison.com/9

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XSIQ

* Intermediate Mathematics - :iscrete probabilit% distribution
http://alison.com/courses/Intermediate-Mathematics-2http://alison.com/courses/Intermediate-Mathematics-2/content/scorm/1609/Discrete-random-variableshttp://alison.com/http://alison.com/courses/Intermediate-Mathematics-2/content/scorm/1609/Discrete-random-variableshttp://alison.com/http://alison.com/courses/Intermediate-Mathematics-2


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Discrete probability distribution

/he discrete probabilit% distribution of a random variable is #iven b% its possible

values to#ether with their probabilities.

/his distribution ma% be displa%ed as a table or a #raph or sometimes" as a

formula.

!irst list all the possible values" x" that Xcan tae.

/hen calculate the associated probabilities" Pr(X = x).

Remember that the total probabilit% ; 1.


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measure o$ spread

Standard deviation as a measure of spread

/wo dice are rolled and the numbers showin# uppermost are added.

7et / be the random variable denotin# the result of one roll. a.

ach die has the numbers 1" " 3" " ," and .

/herefore the value for / mi#ht be the total or the total 3 or the total and so

on./he total 3 mi#ht occur from throws of 1 and or and 1.

/he total of mi#ht occur from throws of 1 and 3" and " and 3 and 1.

A list of the full ran#e of possibilities for the var%in# totals would be2 2 $1" 1) 32

$1" )" $" 1) 2 $1" 3)" $" )" $3" 1) ,2 $1" )" $" 3)" $3" )" $" 1) 2 $1" ,)" $" )"

$3" 3)" $" )" $," 1) 2 $1" )" $" ,)" $3" )" $" 3)" $," )" $" 1) '2 $" )" $3" ,)" $"

)" $," 3)" $" ) +2 $3" )" $" ,)" $," )" $" 3) 1(2 $" )" $," ,)" $" ) 112 $" ,)" $,"

) 12 $" )

/here is a total of 3 possible combinations for the two numbers.

The probability distribution table could be listed as follows:

/he mean mi#ht be calculated with the formula2

/he smallest possible value e?uals 3.

/he lar#est possible value e?uals 11.

In this case" all the values" except for and 1" lie within two standard

deviations of the mean.

/his proportion of the possible values represents +@ of the values.

/hus +@ of the possible values lie within two standard deviations of the mean.

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XSIQ * Intermediate Mathematics - The normal curve The normal curve A series


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of tests was given to a large number of students and the results of these tests

were combined to obtain a Verbal Aptitude score for each student !or the whole

group it was found that" Mean # $%$& and Standard 'eviation # () Samples of

increasing sie +,& $& .& and $/ were ta0en and the scores for each

sample graphed as a histogram As the sample sie increased the distribution

became more s1mmetrical and approached the shape of the normal curve

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--------------------------------------------------------------------------------XSIQ * Intermediate Mathematics - 6ontinuous random variables and the

normal distribution 6ontinuous random variables and the normal distribution

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BIC * Intermediate Mathematics - Dalculation of probabilities for a normal

distribution Dalculation of probabilities for a normal distribution /he E6ormal

distribution - cdfE table can be used in reverse to find the value of the random

variable for a #iven probabilit% or area under the curve. /he name #iven to this

value of the random variable is a Cuantile. /he value of ? will correspond to an

area e?ual to (. under the 6ormal distribution #raph. /he probabilit% value

(. must be found in the Fbod%F of the E6ormal distribution - cdfE table. !irst"

%ou loo for a value close to" but sli#htl% smaller than" the re?uired value. /he

appropriate section of the #eneral 6ormal distribution - cdf table is shown

below. =x= 3 , 1 3 (. (.3, (.3'+ (. (., 3 1( 13 (.

(.3 (.(3 (.3 (. 3 + 1 (.' (.+ (.++, (.'(3 (.'(,1 3 '11 6otice that the probabilit% value (.3 in the bod% of the 6ormal

distribution - cdf table is sli#htl% smaller than the re?uired value (.. /he

probabilit% value (.3 is at the intersection of the row for x ; (. $first

decimal place) and the column for x ; (.(, $second decimal place). /hat is" the

#eneral x value e?uals (., for the probabilit% value (.3. 6otice that the

probabilit% value (.(( in the bod% of the 6ormal distribution - cdf table is in the


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column for x ; (.(( $third decimal place). Ghen we add these two readin#s

to#ether for our case we find that the #eneral x value e?uals (., for the

probabilit% value (.( $(.3 H (.(((). 4revious 5 6ext

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BIC * Intermediate Mathematics - Approximatin# the binomial distribution with

normal distribution Approximatin# the binomial distribution with normal

distribution 4revious 5 6ext

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Mathematics Upper Secondary4