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

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:

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

• What is $Th$? Are $a$ and $b$ independent? – Angina Seng Jan 23 '18 at 4:56
• but are they independent? – Frank Moses Jan 23 '18 at 5:31
• if they are independent then what you wrote is right because the probability $P(a>th)$ will be a constant and its expectation is also that constant – Frank Moses Jan 23 '18 at 5:32
• if $a$ and $b$ are not independent then you might have to use conditional probability relations – Frank Moses Jan 23 '18 at 5:33
• then they should be functions of some channel gains. Right? we need to know what is true about those channel gains – Frank Moses Jan 23 '18 at 5:48

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.