The lifetime of some batteries is exponentially distributed with parameter $\alpha$ if made by company A, and $\beta$ if made by company B. You have an unlabeled box of batteries, and wish to know which company it is from. Assume a probability p = 1/5 that the box is from company B.
The maximum a posteriori (MAP) rule determines in favor of company A, if the test battery has a lifetime greater than or equal to $\theta$.
The probability of error of this MAP estimation is calculated as follows:
$\mathbf P(\textrm{error}) = \mathbf P(\textrm{error}\mid A)\mathbf P(A) + \mathbf P(\textrm{error}\mid B)\mathbf P(B)$
$=\mathbf P(T\geq\theta\mid\ A)(1-p) + \mathbf P(T<\theta\mid B)p$
$=e^{-\alpha\theta}\cdot\frac{1}{5}+(1-e^{-\beta\theta})\cdot\frac{4}{5}$
Can someone explain to me how we have the following?
$$\mathbf P(T\geq\theta\mid\ A)=e^{-\alpha\theta}$$
$$\mathbf P(T<\theta\mid B)=1-e^{-\beta\theta}$$
Thank you so much!!!
The pdf of exponential law of parameter $\lambda$ is given by $f(x) = \lambda e^{-\lambda x}$ (for $x>0$).
Then (skipping the integral calculations),
$$\mathbb{P}(x \geq \theta) = \int_\theta^\infty f(x)dx = e^{-\lambda \theta}$$
and
$$\mathbb{P}(x \leq \theta) = \int_0^\theta f(x)dx = 1- e^{-\lambda \theta}$$
The parameter then change to $\alpha$ or $\beta$ as your are assuming that the data is coming from either company $A$ or $B$.