1. Suppose people arrive at a bank with poisson rate \(\lambda = 4\) per hour.
    1. What is the probability that 5 people arrive in the first half hour?
    2. What is the probability that at least 3 people arrive in the first hour?

  2. Patients arrive at the doctor’s office according to Poisson distribution with \(\lambda = 4\)/hour.

    1. What is the probability of getting less than or equal to 8 patients within 2 hours?
    2. Suppose each arriving patient has 25% chance to bring a person to accompany. There are 20 seats in the waiting room. At least many hours should pass that there is at least 50% probability that the waiting room is filled with patients and their relatives?

  3. Suppose the the pdf of a random variable \(x\) is \(f(x) = \dfrac{a}{(1-x)^{1/3}}\) for \(0 < x < 2\) and \(0\) for other values of x.

    1. Find the constant \(a\).
    2. Find cdf of F(X < 3/4).

  4. Let \(X\) and \(Y\) be the random variables and \(f(x,y)\) is the probability density function of the joint distribution. Suppose \(f(x,y) = a(\dfrac{5x}{7} + \dfrac{9y^3}{2})\) if \(0<x<2\) and \(-1<y<1\) (0 otherwise).

    1. Find \(a\).
    2. Find the marginal distribution of \(y\) (\(h(y)\)) and \(h(y<0.5)\).
    3. Find the conditional distribution of \(f(y|x)\).