What is the simplified form for $Pr [ min (a,b) > Th]$?

Question

As the subject mentions, how can I find simplified form $$P = Pr [ min (a,b) > Th]$$

Note that both $a$ and $b$ are also random variables denoting the SINR of a wireless signal. $Th$ is the threshold value. Both $a$ and $b$ are independent.

Does it mean I have to find the Expected value of $P$? If yes, in this case, what can be the expected value of P

EDIT: Can I say the following:

$$ P = E[ P(min (a,b) > Th)] = E[P(a>Th)].E[P(b>Th)] \ \forall (0<{a,b} < 1)$$

Answer 1

I may write the probability in a different way.

If $min(a,b)$ is greater than $Th$, then it means both $a$ and $b$ are greater than $Th$, so the probability may be rewritten as

$$ P[min(a,b)>Th] = P[ (a > Th) \cap ( b>Th)]$$

The events $min(a,b)>Th$ and $(a>Th) \cap (b>Th)$ are the same.

• As an addition : this concept is quite known in Actuarial Mathematics, in terms of first-failure and last-survivor

Is this okay..?