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Variance of a Random Variable
Dennis Sun
The “Spread” of a Random Variable
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Two random variables can have the same expected value and still
be very different.
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How do we quantify the spread of a random variable?
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Variance of a Random Variable
Recall that the variance of a list of numbers is
Pn
(yi − y)2
2
s = i=1
.
n
We can also write this as a weighted average:
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s2 =
(x − y)2 · wx .
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We can define the variance of a random variable in the same way:
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Var[X] =
(x − E[X])2 · p[x].
x
Notice that the variance cannot be negative.
Standard Deviation of a Random Variable
The variance is difficult to interpret because it is in units2 .
To convert back to the original units, we take the square root:
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SD[X] = Var[X].
The square root of the variance is called the standard deviation.
Example
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Let’s calculate SD
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number
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of tails
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.03125
.15625
.3125
.3125
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.03125