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Using the fact the expected value operator E(X) is linear we have that
`E[g(X) ] = g(E[X])`
if `g(x)` is a linear function.
In this case we are given that `g(x) = a` so that
`E[g(X)] = g(E[X]) = a`
This can be demonstrated using the definition of expectation:
`E[g(X)] = int g(x) f(x) dx`
where `f(x)` is the probability density function of the random variable `x`.
If `g(x) = a` then
`E[g(X)] = int a f(x) dx = a int f(x) dx` (since the constant `a` can be taken outside the integral.
Now, as `f(x)` is a probability density function, we must have that it integrates to` ` 1 so that
`int f(x) dx = 1`
Therefore, `E[g(X)] = a . 1 = a`
`E[g(X)] = a` if `g(x) = a`
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