# Expected value for light bulb that had burned out

In the following example for EM

The equation for the expected value of a light bulb that has burned out is introduced:

$$\theta - \frac{te^{-t/\theta ^{(k)}}}{1-e^{-t/\theta ^{(k)}} }$$

The expression is for $E[X|X<t],$ where $X$ has an exponential distribution with mean $\theta$ and $t>0.$ You can find it using this formula: $$E \left[ X | X < t \right] = \frac{\int_0^t xf(x) dx}{P \left[ X<t \right]}$$