# Probability

Convergence in probability.

For each of the following sequences, determine whether it converges in probability to a constant. If it does, enter the value of the limit. If it does not, enter the number “999".

1) Let X1, X2,… be independent continuous random variables, each uniformly distributed between −1 and 1.

a) Let Ui=X1+X2+⋯+Xii, i=1,2,…. What value does the sequence Ui converge to in probability? (If it does not converge, enter the number “999". Similarly in all below.)

b) Let Σi=X1+X2+⋯+Xi, i=1,2,…. What value does the sequence Σi converge to in probability?

c) Let Ii=1 if Xi≥1/2, and Ii=0, otherwise. Define,

Si=(I1+I2+⋯+Ii)/i.

What value does the sequence Si converge to, in probability?

d) Let Wi=max{X1,…,Xi}, i=1,2,…. What value does the sequence Wi converge to in probability?

e) Let Vi=X1⋅X2⋯Xi, i=1,2,…. What value does the sequence Vi converge to in probability?

2) Let X1, X2,…, be independent identically distributed random variables with E[Xi]=2 and Var(Xi)=9, and let Yi=Xi/2i.

a) What value does the sequence Yi converge to in probability?

b) Let An=1n∑i=1nYi. What value does the sequence An converge to in probability?

c) Let Zi=13Xi+23Xi+1 for i=1,2,…, and let Mn=1n∑i=1nZi for n=1,2,…. What value does the sequence Mn converge to in probability?

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1. See Similar Questions below.

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Writeacher
2. You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

You are having a discussion about sequences with your classmate. She insists that the sequence 2, 3, 5, 8, 12 must be either arithmetic or geometric. Is she correct or incorrect? Explain.

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