expected value of a random variable 1 the expected value of a random variable, also known as its...
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EXPECTED VALUE OF A RANDOM VARIABLE
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The expected value of a random variable, also known as its population mean, is the weighted average of its possible values, the weights being the probabilities attached to the values.
Definition of E(X), the expected value of X:
n
iiinn pxpxpxXE
111 ...)(
2
Note that the sum of the probabilities must be unity, so there is no need to divide by the sum of the weights.
EXPECTED VALUE OF A RANDOM VARIABLE
Definition of E(X), the expected value of X:
n
iiinn pxpxpxXE
111 ...)(
3
EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) This sequence shows how the expected value is calculated, first in abstract and then with the random variable defined in the first sequence. We begin by listing the possible values of X.
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) Next we list the probabilities attached to the different possible values of X.
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) Then we define a column in which the values are weighted by the corresponding probabilities.
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) We do this for each value separately.
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EXPECTED VALUE OF A RANDOM VARIABLE
Here we are assuming that n, the number of possible values, is equal to 11, but it could be any number.
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X)
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EXPECTED VALUE OF A RANDOM VARIABLE
The expected value is the sum of the entries in the third column.
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X)
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) The random variable X defined in the previous sequence could be any of the integers from 2 to 12 with probabilities as shown.
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) X could be equal to 2 with probability 1/36, so the first entry in the calculation of the expected value is 2/36.
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EXPECTED VALUE OF A RANDOM VARIABLE
The probability of x being equal to 3 was 2/36, so the second entry is 6/36.
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X)
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EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) Similarly for the other 9 possible values.
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) 252/36
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To obtain the expected value, we sum the entries in this column.
EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi
x1 p1 x1 p1 2 1/36 2/36
x2 p2 x2 p2 3 2/36 6/36
x3 p3 x3 p3 4 3/36 12/36
x4 p4 x4 p4 5 4/36 20/36
x5 p5 x5 p5 6 5/36 30/36
x6 p6 x6 p6 7 6/36 42/36
x7 p7 x7 p7 8 5/36 40/36
x8 p8 x8 p8 9 4/36 36/36
x9 p9 x9 p9 10 3/36 30/36
x10 p10 x10 p10 11 2/36 22/36
x11 p11 x11 p11 12 1/36 12/36
xi pi = E(X) 252/36 = 7
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EXPECTED VALUE OF A RANDOM VARIABLE
The expected value turns out to be 7. Actually, this was obvious anyway. We saw in the previous sequence that the distribution is symmetrical about 7.
Very often the expected value of a random variable is represented by , the Greek m. If there is more than one random variable, their expected values are differentiated by adding subscripts to .
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EXPECTED VALUE OF A RANDOM VARIABLE
Alternative notation for E(X):
X
n
iiinn pxpxpxXE
111 ...)(
Copyright Christopher Dougherty 2012.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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2012.10.29
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