Probability-Kolmogorov's 0-1 Law (ISS Statistics Paper I (New 2016 MCQ Pattern)): Questions 1 - 3 of 3

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Question number: 1

» Probability » Kolmogorov's 0-1 Law

MCQ▾

Question

Consider the random graph we get from the square grid Z2 by keeping each edge with probability p (0 < p < 1 some predetermined constant) IID, and deleting it otherwise (this is percolation on the square grid, and has been extensively studied). Which of the following is correct regarding an infinite connected component somewhere in the graph?

Choices

Choice (4) Response
a.

There always exist finite connected component.

b.

Existence of an infinite connected component is impossible.

c.

Probability that there exists an infinite connected component is 0.5.

d. None of the above

Question number: 2

» Probability » Kolmogorov's 0-1 Law

MCQ▾

Question

Consider an infinite random stream of fair coin tosses: an infinite sequence of 0′s and 1′s, chosen IID with equal probabilities at each step. Call the result of the ith coin toss Xi (these variables generate the sample space). Which of the following is correct regarding the following limit?

limn(X1++Xn)n

Choices

Choice (4) Response
a.

Strong law of large numbers states that the limit exists, with probability 0.

b.

According to Kolmogorov’s 0 - 1 law the limit never exist.

c.

According to Kolmogorov’s 0 - 1 law if there is no limit then it’s probability is 1.

d. Question does not provide sufficient data or is vague

Question number: 3

» Probability » Kolmogorov's 0-1 Law

MCQ▾

Question

Which of the following is correct regarding Kolmogorov’s zero-one law?

Choices

Choice (4) Response
a.

It is even valid for sequence of dependent random variables.

b.

A tail σ -algebra contains only events of probability 0 or 1.

c. Question does not provide sufficient data or is vague
d. All of the above

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