The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = p^x(1-p)^(1-x).
a) By calculating f(0) and f(1), give a practical example of a Bernoulli experiment, and a Bernoulli random variable.
b) Calculate the mean and variance of the Bernoulli random variable.
2. The geometric random variable X has moment generating function given by E(e^tx) = p(1-qe^t)^-1, where q = 1-p and 0 < p < 1. Use this to derive the mean and variance of X.
3. Let the random variable X be the number of days that a patient needs to be in hospital. Suppose X has p.f. f(x) = k(6-x) x = 1, 2, 3, 4, 5.
a) Find the value of the multiplicative constant k, so that f(x) is a probability function.
b) What is the expected length of stay? If you cannot answer (a), write this in terms of k.
c) Suppose the patient has medical insurance, and they are to receive $200 for each of the first two days in hospital, and $100 for each day after the first two days, what is the expected payout for the hospitalisation? If you cannot answer (a), write this in terms of k.
4. The pdf of X is f(x) = c/x^2, 1 < x < ∞.
a) Calculate the value of c so that f(x) is a pdf.
b) Show that the mean of X does not exist.
c) Interpret the result in (b).
5. The triangular distribution has pdf f(x) = {cx 0 <= x < 1, c(2-x) 1 <= x <= 2.
It is the sum of two independent uniform(0, 1) random variables.
a) Find c so that f(x) is a density function.
b) Draw the pdf, and derive the cdf using simple geometry.
c) Derive the cdf from its definition.
d) Derive the mean and variance of a random variable with this distribution.