Continuous Random Variables: Accept-Reject Method

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In the accept-reject method for generating continuous random variables, what is the purpose of choosing an appropriate proposal distribution?

To determine the value of c for acceptance probability calculations

To ensure the support of the proposal distribution is larger or the same as the support of the target distribution (correct)

To accept or reject samples based on certain probabilities

To match the CDF of the proposal distribution with the target distribution

What is the relationship between the probability of acceptance and the function f(Y)?

Probability of acceptance is inversely proportional to f(Y)

Probability of acceptance is directly proportional to f(Y) (correct)

Probability of acceptance depends on the square of f(Y)

Probability of acceptance is not related to f(Y)

What condition must be satisfied for implementing an accept-reject sampler in continuous random variable generation?

$\sigma_X = \sigma_Y$

$\int_{X} f(x) dx = 1$

$\mathbb{E}(X) = \mathbb{E}(Y)$

$\sup_{x \in X} f(x) \leq c g(x)$ (correct)

In the context of the accept-reject method, what does it mean if f(y) is large but g(y) is small?

The value will be accepted with a high probability Signup and view all the answers

What distribution describes the number of attempts it takes to generate an acceptance in the accept-reject method?

Geometric distribution Signup and view all the answers

What step in Algorithm 1 of the accept-reject method involves drawing a proposal sample from the proposal distribution?

Step 2 Signup and view all the answers

How does Theorem 1 prove that Algorithm 1 returns samples from the target distribution in the accept-reject method?

By showing that the CDF of the generated samples equals the target CDF Signup and view all the answers

How is the Mean number of loops for an acceptance related to the parameter 'c' in the accept-reject method?

Mean number is directly proportional to c Signup and view all the answers

What is essential for showing that Algorithm 1 generates samples from the target distribution in the accept-reject method?

Equality in CDFs between generated samples and target distribution Signup and view all the answers

What does it mean when f(y) is small but g(y) is large in the context of the accept-reject method?

The value will not be proposed often and is likely to be rejected Signup and view all the answers

How is the probability of acceptance related to the Cumulative Distribution Function (CDF) F(x)?

Probability of acceptance equals F(x) Signup and view all the answers

How does the accept-reject method ensure that generated samples follow the desired target distribution?

By adjusting proposal distribution support based on target distribution support Signup and view all the answers

What is the proposal distribution used in the accept-reject method for the Beta(4, 3) distribution?

Uniform distribution Signup and view all the answers

What is the formula for the beta distribution Beta(4, 3) mentioned in the text?

$f(x) = x^4(1 - x)^3$ Signup and view all the answers

What is the critical point for the maximum of the beta distribution Beta(4, 3)?

$x = 3/5$ Signup and view all the answers

In the context of the accept-reject method for Beta(4, 3), what condition leads to accepting a sample?

$U \leq f(Y) / (cg(Y))$ Signup and view all the answers

What is the maximum of the ratio between the probability density function and the proposal distribution for the Beta(4, 3) distribution?

2.5 Signup and view all the answers

When using the accept-reject method for Beta(4, 3), what role does 'c' play in determining whether to accept a sample?

'c' controls the probability of acceptance Signup and view all the answers