2.4 Expectation, Mean and Variance (2)

Mean and Variance of Some Common Random Variables

the mean and the variance of a few important random variables

 

 

Bernoulli

The mean, second moment, and variance of X are given by the following calculations:

 

$$E[X] = 1 \cdot p + 0 \cdot (1 - p) = p,$$

$$E[X2] = 1^2 \cdot P + 0 \cdot (1 - p) = p,$$

$$var(X) = E[X^2] - (E[X])^2 = p - p^2 = p(1 - p).$$

 

 

 

Discrete Uniform Random Variable

 

= discrete uniformly distributed random variable

takes one out of a range of contiguous integer values, with equal probability.

Discrete Uniform Random Variable

 

Its PMF is

 

where a and b are two integers with a < b.

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The mean is

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To calculate the variance of X, we first consider the simpler case where a = 1 and b = n.

 

$$\begin{align*}
var(X) &= E[(X - E[X])^2]\\
 &= \frac{1}{6}(n+1)(2n+1) - (\frac{n+1}{2})^2\\
 &= \frac{1}{12}(n+1)(4n+2-3n-3)\\
 &= \frac{n^2 - 1}{12}
\end{align*}$$

 

 

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A random variable which is uniformly distributed over the interval [a, b] has the same variance as one which is uniformly distributed over [1, b - a + 1], since the PMF of the second is just a shifted version of the PMF of the first.

 

The above formula with n = b - a + 1,

 

 

 

 

The Mean of the Poisson

 

Poisson PMF

can be calculated is follows:

 

The last equality is obtained by noting that

is the normalization property for the Poisson PMF.

 

Note: the variance of a Poisson random variable is also \(\lambda\)

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Decision Making Using Expected Values

Expected values often provide a convenient vehicle for optimizing the choice between several candidate decisions that result in random rewards.

If we view the expected reward of a decision as its "average payoff over a large number of trials," it is reasonable to choose a decision with maximum expected reward.

 

 

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References

Bertsekas, D. P., Tsitsiklis, J. N. (2008). Introduction to Probability Second Edition. Athena Scientific.