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 doctorâ€™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 $$f(y|x)$$.