Question

Let \(X_1,X_2,X_3\) be a random sample of size \(3\) from \(U(1,2)\) distribution. Then \(P(X_{(3)}>1.5)\) equals (rounded off to two decimal places).

Hint:

For the maximum order statistic \(X_{(n)}\), use \[ P(X_{(n)}\leq x)=\left(F(x)\right)^n \] when the observations are independent and identically distributed.

Answer 1

Correct Answer: 0.88

Step 1: Read the event.
$X_{(3)}$ is the largest of the three, so $X_{(3)}>1.5$ means at least one value exceeds $1.5$.

Step 2: Use the complement.
$P(X_{(3)}>1.5)=1-P(\text{all three}\le1.5)$.

Step 3: Single value probability.
For $U(1,2)$, $P(X\le1.5)=\frac{1.5-1}{1}=0.5$.

Step 4: Combine.
By independence $P(\text{all}\le1.5)=0.5^3=0.125$, so the answer is $1-0.125=0.875$.

Step 5: Round.
\[ \boxed{0.88} \]