Probability problems day 10

  1. The time between calls at a call center follows an exponential distribution with average value 5 minutes. What is the probability that there are 10 minutes with no call?

  2. Let \(U\) be uniform on \([-1, 1]\).

    1. Compute \(E[U]\), \(V(U)\), and \(E[U^n]\) for any \(n \geq 0\).
    2. Is \(U^2\) uniform on \([0, 1]\)? What is its pdf?

  3. Let \(X_1\), \(X_2\), and \(X_3\) be three independent exponential distributions with mean \(1\). Let \[X = \max(X_1, X_2, X_3)\] be the largest of the three. What is the pdf of \(X\)?

    Hint. \(X \leq x\) if and only if \(X_1 \leq x, X_2 \leq x, X_3 \leq x\).