Probability of i.i.d uniform random variables

Question

I have the following problem and I do not know how to proceed:

Let $0<s<t$ and let $U_{1}$ and $U_{2}$ be i.i.d uniform[0;t]. Compute: $P(U_{1}>s;U_{2}>s)$

Any help is useful. Thanks.

Answer 1

i.i.d. means "independent and identically distributed".

Independent means $\mathsf P(U_1>s, U_2>s)=\mathsf P(\textsf{what?})\cdot\mathsf P(\textsf{what now?})$

Identically distributed means $\mathsf P(\textsf{what?})=\mathsf P(\textsf{what now?})$

And a uniform distribution means something too.

Your task is to remember what's what and put it together now.

Answer 2

Hint

1. Compute $P(U>s)$ for a $U(0,t)$ using the definition of the uniform distribution.

2. Since they are independent, $P(U_1>s,U_2>s) = P(U_1>s)P(U_2>s) = P(U>s)^2.$