Introduction to Continuous Probability Distributions

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7/21/2019 Introduction to Continuous Probability Distributions
http://slidepdf.com/reader/full/introductiontocontinuousprobabilitydistributions 1/14
Introduction to Continuous Probability Distributions
In this lesson, we will discuss two of these: the uniform distribution and the normal
distribution. The normal distribution is particularly important because many of the
methods used in statistics are based on this distribution
Uniform Distribution
We learned that a continuous random variable has a set of possible values that is an
interval on the number line. It is not possible to assin a probability to each point in
the interval. Instead, the probability distribution of a continuous random variable ! is
specified by a mathematical function f"#$ called the probability density function or
%ust density function. The raph of a density function is a smooth curve. & probability
density function "pdf$ must satisfy two conditions: "'$ f"#$ ( ) for all real values of #
and "*$ the total area under the density curve is e+ual to '. The raphs of three density
functions are shown in iure ''.'.
The probability that ! lies in any particular interval is shown by the area under the
density curve and above the interval. The followin three events are fre+uently
encountered: "'$ !  a, the event that the random variable ! assumes a value less than
a "*$ a  !  b, the event that the random variable ! assumes a value between a and
b and "/$ ! 0 b, the event that the random variable ! is reater than b. We say that
we are interested in the lower tail probability for "'$ and the upper tail probability
when usin "/$. The areas associated with each of these are shown in iure ''.*.
1otice that the probability that a  !  b may be computed usin tail probabilities:
P"a  !  b$ 2 P"!  b$ 3 P"!  a$.
If the random variable ! is e+ually li4ely to assume any value in an interval "a, b$,
then ! is a uniform random variable. The pdf is flat and is above the #5a#is between a
and b, and it is ) outside of the interval. The heiht of the curve must be such that the
area under the density and above the #5a#is is '. 6ecause this reion is a rectanle, the
area is the heiht times the width of the interval, which is b 3 a. Thus, the heiht must
be that is, the pdf of a uniform random variable has the form
2 ), otherwise.& raph of the pdf is shown in iure ''./.
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7#ample
& roup of volcanoloists "people who study volcanoes$ has been monitorin a
volcano8s seismicity, or the fre+uency and distribution of underlyin earth+ua4es.
6ased on these readins, they believe that the volcano will erupt within the ne#t *9
hours, but the eruption is e+ually li4ely to occur any time within that period. What isthe probability that it will erupt within the ne#t eiht hours
;olution
Define ! 2 the time until the eruption of the volcano. ! has positive probability over
the interval "),*9$ because the volcano will erupt durin that time interval. 6ecause
the lenth of the interval is *9 3 ) 2 *9, the heiht of the density curve must be for
the area under the density and above the #5a#is to be one. That is, the pdf is
2 ), otherwise.
The probability that the volcano will erupt within the ne#t eiht hours is e+ual to the
area under the curve and above the interval "),<$ as shown in iure ''.9. This area is
.
*
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In the previous e#ample, notice that the area is the same whether we have P")  ! 
<$ or P") = !  <$ or P")  ! = <$ or P") = ! = <$. Unli4e discrete random variables,
whether the ine+uality is strict or not, the probability is the same for the continuous
random variables. This also correctly implies that, for continuous random variables,
the probability that the random variable e+uals a specific value is ).
Normal Distribution
Normal Probability Distributions
1ormal probability distributions are continuous probability distributions that are bell
shaped and symmetric. They are also 4nown as >aussian distributions or bell5shaped
curves.
The normal distribution is perhaps the most widely used probability distribution,
larely because it provides a reasonable appro#imation to the distribution of many
random variables. It also plays a central role in many of the statistical methods that
will be discussed in later lessons. 1ormal probability distributions are continuous
probability distributions that are bell shaped and symmetric as displayed in iure
''.?. The distribution is also called the >aussian distribution or the bell5shaped curve.
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The normal distribution has two parameters: the mean @ and the standard deviation A.The notation ! B 1"@ ,A$ means that ! is normally distributed with a mean of @ and
a standard deviation of A. The distribution is symmetric about the mean. The mean,
median, and mode are all e+ual. The mean is often referred to as the location
parameter because it determines where the distribution is centered. The standard
deviation determines the spread of the distribution. The effect of the mean and
standard deviation on the normal distribution is displayed in iure ''..
or any normal distribution, about <E of the observations are within one standard
deviation of the mean. &bout F?E and FF.GE of the observations are, respectively,
within two and three standard deviations of the mean.
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It is important to remember that, althouh the location and spread may chane, the
area under the curve and above the #5a#is is always '. Unfortunately, the probabilities
associated with intervals cannot be computed easily as with the uniform distribution.
To overcome this difficulty, we rely on a table of areas for a reference of normal
distribution called the standard normal distribution. The standard normal distribution
is the normal distribution with @ 2 ) and A 2 '. It is customary to use the letter H torepresent a standard normal random variable.
We will first learn to compute probabilities for a standard normal random variable and
then learn how to find them for any random variable. We will also want to be able to
determine e#treme values of H, such as the value that only ?E of the population
e#ceeds or the value that 'E of the population is less than. To find either probabilities
or e#treme values, we need a table of standard normal curve areas, or we need a
calculator or computer that can be used to find these values. ere, we will restrict
ourselves to the use of tables. The standard normal table used here in Table ''.'
tabulates the probability of observin a value less than or e+ual to H "see iure ''.G$.
?
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>raphs are e#tremely useful tools to help us understand what values we are searchin
for. We will do this for each problem we wor4.
7#amples of Continuous Probability Distributions
6elow are nine e#amples of continuous probability distributions problems and
solutions.
7#ample '
ind P"H  './*$.
;olution '
Usin the standard normal table, we find the row with './ in the H column and move
alon that row to the ).)* column to find ).F). Thus, P"H  './*$ 2 ).F). iure
''.< shows the raphic imae of this.
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7#ample *
ind P"H 0 './*$.
;olution *
rom the table, we find P"H  './*$ as we did in the previous e#ample. Usin some of
the ideas of probability we learned earlier, we have P"H 0 )./*$ 2 ' 3P"H = './*$ 2 ' 3
).F) 2 ).)F/9. ;ee iure ''.F.
7#ample /
ind P"H  3).?$.
;olution /
There are no neative H5values in the table, so we cannot loo4 this up directly. Instead,
we use the symmetry of the normal distribution to find the probability "see iure
''.')$. That is,
P"H  3).?$ 2P"H 0 ).?$
2 ' 3 P"H  ).?$
2 ' 3 ).F'?
2 )./)<?
<
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7#ample 9
ind P"3'.9?  H  ).G$.
;olution 9
iure ''.'' shows the solution.
irst, we notice that P"3'.9?  H  ).G$ 2 P"H  ).G$ 3 P"H  3'.9?$. 1ow P"H ).G$ can be found directly from the table to be ).GG9. Usin the symmetry of the
normal distribution aain, P"H  3'.9?$ 2 P"H 0 '.9?$ 2 ' 3 P"H = '.9?$ 2 ' 3 ).F*? 2
).)G/?. inally, P"3'.9?  H  ).G$ 2 P"H  ).G$ 3 P"H  3'.9?$ 2 ).GG9 3 ).)G/? 2
).G)*F.
7#ample ?
ind the value HJ such that P"H  HJ$ 2 ).G?.
;olution ?
F
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This is different from the other problems we have considered. Instead of findin a
probability, we are loo4in for a H5value. owever, the same table will allow us to
solve the problem. The difference is that we will loo4 in the table for a probability and
then find the H5value associated with the probability. Koo4in in the body of the table,
we find the values ).G9< and ).G?'G, which are the closest to the ).G? of interest. 6y
loo4in at the correspondin row and column headins, we find that P"H  ).G$ 2).G9< and P"H  3).<$ 2 ).G?'G. 6ecause ).G9< is closer to ).G? than ).G?'G, we
ta4e HJ 2 ).G. "1ote: We could interpolate to find a more precise value of HJ, but we
will not o throuh this process here.$ ;ee iure ''.'*.
7#ample
ind the value HJ such that P"H 0 HJ$ 2 ).)?.
;olution
We need to have the probabilities in the form P"H  HJ$ to use the table. owever, P"H
0 HJ$ 2 ' 3 P"H = HJ$. We can rewrite this as P"H = HJ$ 2 ' 3 P"H 0 HJ$ 2 ' 3 ).)? 2
).F?. That is, if ?E of the population values are reater than HJ, then F?E of the
population values must be less than or e+ual to HJ. Thus, we loo4 for ).F? in the body
of the table and find ).F9F? and ).F?)? correspondin to H 2 '.9 and H 2 '.?,
respectively. 6ecause ).F? is e#actly halfway between ).F9F? and ).F?)?, we have HJ2 '.9?. "This is the only time we don8t %ust round to the closest value.$ ;ee iure
''.'/.
')
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7#ample G
ind the value HJ such that P"H  HJ$ 2 ).)'.
;olution G
6ecause the standard normal is symmetric about its mean ), we 4now P"H  )$ 2 ).?,
we 4now that HJ must be less than ). &lso, because we have only positive values of H
in the table, we cannot loo4 for ).)' directly in the table. owever, aain because of
symmetry, we 4now that, if P"H  HJ$ 2 ).)', then P"H 0 3HJ$ 2 ).)'. To use the table,
we must find P"H= 3HJ$ 2' 3 P"H 0 3HJ$ 2 ' 3 ).)' 2 ).FF. Koo4in in the body of the
table, we find ).F<F< and ).FF)', correspondin to H 2 *./* and H 2 *.//,
respectively, to be the closest to ).FF. 6ecause ).FF)' is the closer of the two to *.//,
we find HJ 2 3*.//. ;ee iure ''.'9.
ew normal random variables actually have a standard normal distribution. owever,
any normal random variable can be transformed to a standard normal, and anystandard normal random variable can be transformed to a normal random variable
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with any mean @ and standard deviation A. ;pecifically, if ! B 1"@,A$$,
. urther, if H B 1"),'$, then ! 2 @ L AH B 1"@,A$. Usin these
relationships, we can find probabilities and e#treme values for any normal random
variable usin the H5table. When doin this, it is important to do all calculations
carefully.
7#ample <
Ket ! B 1"'),?$. ind P"!  '?$.
;olution <
P"!  '?$
2 P"H  '$
2 ).<9'/.
1otice that inside the parentheses, we had to transform both the ! and the '? to avoid
chanin the ine+uality. When wor4in with !, we used symbols, and we used
numbers when wor4in with '?. owever, we used the numbers that were associated
with each symbol. Mnce we have the e#pression in terms of H, then the problem is
e+uivalent to the earlier ones we wor4ed. ;ee iure ''.'?.
7#ample F
Ket !, 1"?,*$. ind !J such that P"! 0 !J$ 2 ).)?.
;olution F
'*
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irst, we find HJ such that P"H 0 HJ$ 2 ).)?. rom our earlier wor4, we 4now that HJ 2
'.9?. Then !J 2 @ ' AHJ 2 L *"'.9?$ 2 <.*F.
Continuous Probabiity Distributions In ;hort
We have discussed two continuous distributions: the uniform and the normal. When
every value in an interval is e+ually li4ely to occur, then we have a uniform
distribution. The normal distribution is the most commonly used continuous
distribution. Probabilities associated with a normal random variable must be found by
usin tables, calculators, or computers. When usin tables, it is possible to use only
one table. 6y tradition, the probabilities of a standard normal distribution are
tabulated. Probabilities for other normal random variables are found by transformin
the problem to one on the standard normal.
Notes:
Continuous Probability Distributions
a random variable that can assume all possible random values "ie city
temperature$
Probability Density Function: a function that describes how li4ely
this random variable will occur at a iven point.
Height formula: heiht 2 'N"b5a$ where b is the top rane, and a is the
bottom rane iven.
The Normal Distribution
used to solve continuous probabilities
symmetry about the mean
total area under the curve is '
standard deviation is the distance from the mean to the point of
inflection
&ny normal distribution can be described as by the mean and the
variance: so we often write 1"mean, variance$ to describe a distribution
The distribution chart shows area under the raph from the ! value tothe left end
'/
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ZScores can be calculated usin 1ormal distributions
O 2 # 3 mean N standard deviation
;ometimes, you will have to subtract the mean to e+ualiHe. This
ma4es it so the mean is on the center.
'9