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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

    -

    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

    -

    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.. . . !