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Ludmilka [50]
3 years ago
13

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+⋯+Iii.
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?

Mathematics
1 answer:
nikklg [1K]3 years ago
6 0

Answer:

Check the explanation

Step-by-step explanation:

Kindly check the attached images below for the step by step explanation to the question above.

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Answer:

Option D)Neither solution is extraneous.

Step-by-step explanation:

we have

\sqrt{11-2x}=\sqrt{x^{2}+4x+4}

we know that

two possible solutions are x=-7 and x=1

<u><em>Verify each solution</em></u>

Substitute each value of x in the expression above and interpret the results

1) For x=-7

\sqrt{11-2(-7)}=\sqrt{-7^{2}+4(-7)+4}

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therefore

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2) For x=1

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therefore

x=1 is not a an extraneous solution

therefore

Neither solution is extraneous

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