Virtual Labs

Tools

Aim

Theory

Pretest

Procedure

Experiments

Posttest

References

Contributors

Feedback

Aim

Theory

Pretest

Procedure

Experiments

Posttest

References

Contributors

Feedback

Convergence of Random Variables

Choose difficulty:

Beginner

Intermediate

Advanced

1: Which of the following implies convergence in distribution?

a: Convergence in mean square Explanation

Explanation

b: Convergence in probability Explanation

Explanation

c: Convergence in variance Explanation

Explanation

d: Convergence almost surely Explanation

Explanation

2: Which of the following best defines convergence in probability?

\mathbb{P}(X_n = X) \to 1

Explanation

\mathbb{P}(|X_n - X| > \epsilon) \to 0

for every

\epsilon > 0

Explanation

\mathbb{E}[|X_n - X|^p] \to 0

Explanation

d: The CDF of

X_n

converges pointwise to the CDF of

X

Explanation

3: Which type of convergence is strongest among the following?

a: Convergence in distribution Explanation

Explanation

b: Convergence in probability Explanation

Explanation

c: Convergence almost sure Explanation

Explanation

d: Convergence in Mean Square Explanation

Explanation

4: Let

F_{X_n}(x) = \begin{cases} 1 - \left(1 - \frac{1}{n}\right)^{nx}, & x > 0 \ 0, & x \leq 0 \end{cases}

. What does

X_n

converge in distribution to?

a: U(0,1) Explanation

Explanation

b: N(0,1) Explanation

Explanation

c: Exp(1) Explanation

Explanation

d: Pois(1) Explanation

Explanation

5: What does convergence almost surely mean?

a: The probability that

X_n = X

eventually is 1 Explanation

Explanation

b: The average error

|X_n - X|

goes to zero Explanation

Explanation

c: The values of

X_n

get arbitrarily close to

X

for almost all outcomes Explanation

Explanation

d: The distribution of

X_n

matches that of

X

Explanation

Community Links Sakshat Portal Outreach Portal FAQ: Virtual Labs

AGPL 3.0 & Creative Commons (CC BY-NC-SA 4.0)