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Theory

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Procedure

Experiments

Posttest

References

Contributors

Feedback

Convergence of Random Variables

Choose difficulty:

Beginner

Intermediate

Advanced

1: Which of the following is a necessary condition for the Weak Law of Large Numbers to hold?

a: Random variables must be bounded Explanation

Explanation

b: Random variables must have finite variance Explanation

Explanation

c: Random variables must have a finite mean Explanation

Explanation

d: Random variables must be deterministic Explanation

Explanation

2: Which type of convergence does the Weak Law of Large Numbers (WLLN) guarantee?

a: Convergence in distribution Explanation

Explanation

b: Convergence almost surely Explanation

Explanation

c: Convergence in probability Explanation

Explanation

d: Convergence in total variation Explanation

Explanation

3: In the experiment with shrinking intervals (moving rectangles), what is the primary reason the sequence does not converge almost surely?

a: The sample mean converges to a wrong value Explanation

Explanation

b: Each point ω lies in infinitely many non-overlapping intervals Explanation

Explanation

c: The expectation does not exist Explanation

Explanation

d: The random variables are not identically distributed Explanation

Explanation

4: Suppose

X_n \xrightarrow{a.s.} X

. Which of the following must be true?

a: There exists a subsequence of

X_n

that converges in mean to

X

Explanation

b: Every subsequence of

X_n

converges in probability to

X

Explanation

X_n

is bounded almost surely Explanation

Explanation

X_n

converges in distribution to a constant Explanation

Explanation

5: Which of the following sequences converges in probability to

0

X_n = 1

with probability

1/n

0

otherwise Explanation

Explanation

X_n = 1

if n is even,

0

if n is odd Explanation

Explanation

X_n \sim

Uniform(0, 1/n)

Explanation

d: Both a and c Explanation

Explanation

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