Let
𝑓
(
𝑥
)
=
𝑠
𝑖
𝑛
(
𝑥
)
, for
𝑥
∈
[
0
,
𝜋
/
2
]
.

Would the following R code generate the random samples of f(x)?

m

Question

Let 
𝑓
(
𝑥
)
=
𝑠
𝑖
𝑛
(
𝑥
)
, for 
𝑥
∈
[
0
,
𝜋
/
2
]
.

Would the following R code generate the random samples of f(x)?

m <- 1000

u <- runif(m)

x <- u * pi / 2-----------------------------------------------------(1)

M <- max(sin(x) / (2 / pi))---------------------------------------(2)

x_accepted <- x[which(u <= sin(x) / (M * (2 / pi)))] ----------(3)

BlackTom AI · Accepted Answer

Here is the question restated for clarity:
Let f(x) = sin(x) for x in [0, pi/2]. Would the following R code generate random samples of f(x)?
m <- 1000
u <- runif(m)
x <- u * pi / 2
M <- max(sin(x) / (2 / pi))
x_accepted <- x[which(u <= sin(x) / (M * (2 / pi)))]

Answer option to analyze:
1) No, there is a mistake in (3)

Now, evaluate each part and option carefully:
- What is the intended sampling method here? The code attempts to perform rejection sampling to sample from the density proportional to sin(x) on [0, pi/2], using a uniform proposal x ~ Uniform(0, pi/2). The acceptance probability should be f(x) / (M g(x)), where f(x) = sin(x) is the target up to a constant, and g(x) is the proposal density. Since the proposal is uniform on [0, pi/2], g(x) = 1 / (pi/2) = 2/pi. Therefore the acceptance criterion should be u <= f(x) / (M g(x)) = sin(x) / (M * (2 / pi)).
- Check (3): The code uses x_accepted <- x[which(u <= sin(x) / (M * (2 / pi)))]. This is algebraically the same as the required criterion, so at first glance (3) itself looks structurally correct given M is defined properly.
- Check (2): Here lies the subtle but crucial issue. M is computed as max(sin(x) / (2 / pi)) over x drawn from the same sample. This quantity is an empirical estimate of the true sup of sin(x)/(2/pi) on [0, pi/2]. The theoretical maximum of sin(x)/(2/pi) over the entire interval is (1) / (2/pi) = pi/2, achieved at x = pi/2. If the random sample x does not include a value very close to pi/2, then max(sin(x) / (2 / pi)) will be less than the true maximum pi/2. Using this underestimated M in the acceptance criterion can yield an acceptance probability greater than 1 for some x (since sin(x)/(M*(2/pi)) could exceed 1 when M is too small), which invalidates the rejection sampling framework and biases the results.
- Therefore, the core mistake is in step (2): M should be the known true upper bound pi/2 (or at least computed with a safeguard to ensure it is not underestimated), not merely the maximum from a finite sample. If one used M = pi/2 explicitly, or ensured M is never below the true maximum, the acceptance condition in (3) would be valid.
- Why option (1) (the provided claim) is incorrect: It asserts that there is a mistake in (3). While (3) can be correct given a proper M, the actual pitfall is in (2). Since (2) yields an underestimated M when derived from a finite sample, the sampling procedure can break the theoretical guarantee of the rejection sampler. Therefore saying the error is in (3) misidentifies the root cause.
- In summary, the misstep is not in the acceptance check itself (3) but in the computation of M in (2). The code as written risks using an too-small M and thus producing biased samples, whereas the fix is to set M to the true upper bound pi/2 (or ensure M is not underestimated).

25F-STATS-102C-LEC-1 Quiz3- Requires Respondus LockDown Browser

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类似问题

Let 𝑓 ( 𝑥 ) = 𝑐 𝑜 𝑠 ( 𝑥 ) , for 𝑥 ∈ [ 0 , 𝜋 / 2 ] . Suppose we want to sample 𝑓 ( 𝑥 ) through the rejection method, with 𝑈 𝑛 𝑖 𝑓 ( 0 , 𝜋 / 2 ) as the candidate density. What is the multiplicative constant M?

Considering the rejection method algorithm, all the required conditions are satisfied with the candidate density 𝑔 ( 𝑥 ) and multiplicative constant M, what is the range of 𝑓 ( 𝑥 ) 𝑀 𝑔 ( 𝑥 ) ?

Which of the following descriptions is correct about the rejection method for sampling from a target density 𝑓 ( 𝑥 ) ?

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