Let X, Y be a continuous random variables having a cumulative distribution function F(x, y). Let...

Transcribed Image Text:

Let X, Y be a continuous random variables having a cumulative distribution function F(x, y). Let their marginal (cumulative) distributions be G(x) and H(y). (a) Show that G(X) is uniformly distributed in (0, 1). (b) Suppose that you have access to a random variable U uniformly distributed in (0, 1) (for example, in MATLAB or C, you will have access to a uniform random variable). How would you use it to simulate a continuous random variable X having a distribution function G? Justify rigorously. (c) Suppose now you have two IID random variables U₁ and U₂ distributed uniformly in (0,1). How would you use them to simulate random variable pair (X, Y) having a joint distribution F(x, y)? Let X, Y be a continuous random variables having a cumulative distribution function F(x, y). Let their marginal (cumulative) distributions be G(x) and H(y). (a) Show that G(X) is uniformly distributed in (0, 1). (b) Suppose that you have access to a random variable U uniformly distributed in (0, 1) (for example, in MATLAB or C, you will have access to a uniform random variable). How would you use it to simulate a continuous random variable X having a distribution function G? Justify rigorously. (c) Suppose now you have two IID random variables U₁ and U₂ distributed uniformly in (0,1). How would you use them to simulate random variable pair (X, Y) having a joint distribution F(x, y)?

Answer rating: 100% (QA)

The detailed ... View the full answer