Question

Sampling random variables. Let ξ be a random variable and F be its c.d.f. For every u ∈ (0, 1) set q(u) = inf F(x) ≥ u. The function q is called the quantile function of ξ. 1. Show that q is well defined. 2. Give q when ξ ∼ Ber(p) and ξ ∼ Uni(0, 1). 3. Let U be a random variable uniformly distributed over (0, 1). Show that q(U) ∼ ξ. 4. Assuming you can run a random variable uniformly distributed over (0, 1), how to generate ξ of law Exp(λ).

Answer 1

1. We can show that q is well defined by using the properties of cumulative distribution functions (CDFs). 2. For a Bernoulli random variable ξ and a uniform random variable ξ, the quantile functions q(u) are given. 3. We can show that q(U) follows the same distribution as ξ by comparing their cumulative distribution functions. 4. To generate a random variable with an exponential distribution, we can use the inverse transform method.

Step-by-step explanation:

1. To show that q is well defined, we need to prove that for any u in the interval (0, 1), there exists a unique x such that F(x) ≥ u. This can be done by using the properties of cumulative distribution functions (CDFs) and the fact that they are non-decreasing.

2. For a random variable ξ following the Bernoulli distribution with parameter p, the quantile function q(u) is given by:

- q(u) = 0 if u < 1-p

- q(u) = 1 if u ≥ 1-p

For a random variable ξ following the uniform distribution between 0 and 1, the quantile function q(u) is given by:

- q(u) = u

3. Let U be a random variable uniformly distributed over (0, 1). To show that q(U) ∼ ξ, we need to prove that q(U) follows the same distribution as ξ, which is the distribution of U itself in this case. This can be done by showing that the CDF of q(U) is equal to the CDF of ξ.

4. To generate a random variable ξ with an exponential distribution Exp(λ), we can use the inverse transform method. Assuming you have access to a random variable U that is uniformly distributed over (0, 1), the quantile function q(u) is given by:

- q(u) = -ln(1-u) / λ