statistical methods in transport analysis
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Statistical Methods inTransport Analysis
Chapter-0 4
MATHAMATICALEXPECTATION
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04.1 _Expected value
Expected Value is another word formean or average.
It gives a general impression of thebehavior of some random variablewithout giving full details of its
probability distribution or probabilitydensity function
Two random variables with the same
expected value can have very different
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Expected value cont
Mean of expectation
Expected value of X
= E(x) = x f(x) - discrete
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x
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Expected value of X cont
Example_01: Tossing a die (discretevariable)
= E(x) = x f(x)
= 1 x 1/6 + 2 x 1/6 + 3 x /6 + 4 x 1/6 + 5 x 1/6 +6 x 1/6
= 3.5
x 1 2 3 4 5 6
P(X=x)
1/6 1/6 1/6 1/6 1/6 1/6
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Example_02: A carnival game
consists of tossing a dart, which lands
at a random spot within the square
target to the right. The red circle wins
$5, the blue circle wins Rs 7 and either
black triangle wins Rs 10. If it costs Rs 5 to playthis game, do you expect to make money or losemoney?
Let R be the payoff:
Value of R Probability Product
5 /16 0.98175
7 /16 1.3744510 1/4 2.50
Total 4.8562
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Example_03:(Infinite Sum Expected Value)Toss a coin until heads appears, and let R be thenumber of tosses required. What is the expected
value of R?
Let S = 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ...
Value of R Probability Product
1 1/2 1/2
2 1/4 2/4
3 1/8 3/8
4 1/16 4/165 1/32 5/32
n 1/2n n/2n
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Mean of expectation
cont Expected value of a function, g(x)
g(x) = E[g(x)] = g(x)f(x) -discrete
g(x) = E[g(x)] = g(x)f(x) -continuous
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x
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Expected value of a
function,g(x) cont
Example_02: The total distance measured in unitsof 100 km, that private bus operates during a week isa continuous random variable X that has the density
function,x 0
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Mean of expectation
cont Expected value of joint
distribution, g(x,y)
= E[g(x,y)] = g(x,y) f(x,y)
= E[g(x,y)] = g(x,y)
f(x,y)dxdy
x y
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Expected value of jointdistribution, g(x,y) cont
F(x,y) = 1/50 (x2+y2) 0
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Properties of Expectations
E(ax+b) = a E(x) + b
= (ax+b)f(x)dx
= a xf(x)dx + b f(x)dx
= a E(x) + b
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Properties of Expectations
E(ax+b) = a E(x) + b
Expectation of two functions
E[g(x) +h(x)] = E[g(x)] + E[h(x)]
E[g(x) - h(x)] = E[g(x)] - E[h(x)]
For two random variables
E[g(x,y) +h(x,y)] = E[g(x,y)] + E[h(x,y)]
E[g(x,y) - h(x,y)] = E[g(x,y)] - E[h(x,y)]
If x and y are independent,
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04.2 _ Variance andCovariance
Although the mean or expected
value describes where the
probability distribution is centered,the variance; non-negative number
which gives an idea of how widely
spread the values of the random
variable are likely to be; the larger
the variance, the more scattered the
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Variance and Covariancecont
Notes
The larger the variance, the further that
individual values of the random variable(observations) tend to be from the mean.
The smaller the variance, the closer thatindividual values of the random variable
(observations) tend to be to the mean. Taking the square root of the variance
gives the standard deviation, S.
The variance and standard deviation of arandom variable are alwa s non-ne ative
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Variance and Covariancecont
2 =E[(x- ) 2]
= (x- ) 2f(x) - discrete
= (x- ) 2f(x) dx - continuous
Variance of g(x) = E[(g(x) - g(x))2]
= [g(x) - g(x)] 2f(x) -discrete
= [g(x) - g(x)] 2f(x)dx -
continuous
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Properties of Variance
2 = E[x2] 2
= E[(x- ) 2]
= E[(x2 2x + 2 )]
= E[x2] 2 E[x] + 2
= E[x2] 2 2+ 2 = E[x2] 2
2ax = a2 2x
2ax+b = 2ax
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Coefficient of Variance
Coefficient of variance =
, have equal units
So the coefficient of variance is adimensionless value
Higher coefficient of variance largespread
Lower coefficient of variancenegligible spread
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Example_03: The probabilitydistribution of the number of cars
arriving at a toll booth during anyminute is as follows. Calculate the,
Expected value? variance? coefficient of variance?
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= x P(x) = 0.99
E(x2) = x2P(x) = 1.95
2 = E[x2] 2= 1.95 0.992
= 0.97 = 0.984
/ = 0.984/0.99 = 0.994 (Large
X P(x) xP(x) x2P(x) (x-)2P(x)
0 0.37 0 0 0.3621 0.37 0.37 0.37 0
2 0.18 0.36 0.72 0.183
3 0.06 0.18 0.54 0.242
4 0.02 0.08 0.32 0.181
sum 1.00 0.99 1.95 0.968
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THANK YOU.. . . !