Let (Tn)n≥1=T1,T2,… be a sequence of r.v.s such that
Tn∼Unif(5−12n,5+12n).
Given an arbitrary fixed number 0<δ<1, find the smallest number N (in terms of δ ) such that P({|Tn−5|>δ})=0 whenever n>N .
N=
unanswered
Does (Tn)n≥1 converge in probability to a constant? If so, what is the limiting value? Enter DNE if ( {Xn} ) does not converge in probability.
(Tn)n≥1⟶P
unanswered
Does (Tn)n≥1 converge in distribution?
Yes
No
unanswered
Let Fn(t) be the cdf of Tn and F(t) be the cdf of the constant limit. For which values of t does limn→∞Fn(t)=F(t) ? (Choose all that apply.)
t<5
t=5
t>5
unanswered
STANDARD NOTATION
Standard Notation
×
Guidelines Example Entries Common mistakes
Symbols These are case sensitive.
Use the correct case as specified in the problem. n and N are different. Do NOT enter x for X
Parentheses Match each open parenthesis with a close parenthesis.
Elementary arithmetic operations Use the symbols +,-,*,/ for addition, subtraction, multiplication, and division, respectively. Enter 1+b*c-d/e for 1+bc−d/e
For multiplication, use * explicitly.
Although the ''pretty'' display underneath your answer looks correct if you do not include * s, your answer will be marked incorrect! Enter b*c for bc in the example above
Enter 2*n*(n+1) for 2n(n+1) Do NOT enter bc for bc
Do NOT enter 2n(n+1) for 2n(n+1)
Exponents Use the symbol ^ to denote exponentiation. Enter 2^n for 2n
Enter x^(n+1) for xn+1
Square root use the string of letters sqrt, followed by enclosing what is under the square root in parentheses. Enter sqrt(-1) for −1−−−√
Mathematical
constants Use the symbol e for the base of the natural logarithm, e .
Use the string of letters pi for π . Enter e^(i*(pi))+1 for eiπ+1
Order of operations 1) parentheses
2) exponents and roots
3) multiplication and division
4) addition and subtraction
When in doubt, use additional parentheses to remove possible ambiguitites. Enter a/b*c for ab⋅c
Enter a/(b*c) for abc
Enter (1/sqrt(2*(pi)))*e^(-(x^2)/2) for 12π−−√e−x22
Do NOT enter a/b*c for abc
Natural logarithm Although in lectures and solved problems we will sometimes use the notation ''log'' (instead of ''ln''), you should use the string of letters ln, followed by the argument enclosed in parentheses. Enter ln(2*x) for ln(2x) Do NOT enter log(2*x) for ln(2x)
Trigonometric
functions Use the usual 3-letter symbols to denote the standard trigonometric functions Enter sin(x) for sin(x) Do NOT enter sin x for sin(x)
Greek letters Use the Latin-character name to denote each Greek letter Enter lambda*e^(-lambda*t) for λe−λt
Enter mu*alpha*theta for muαθ
Factorials, permutations, combinations Enter fact(k) or factorial(k) for k!
Do NOT use ! in your answers as it will not be evaluated correctly!
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Convergence in probability and in distribution 2
4 points possible (graded)
Let (Yn)n≥1 be a sequence of i.i.d. random variables with Yn∼Unif(0,1).
Let
Mn=max(Y1,Y2,…,Yn).
For any fixed number 0<δ<1 , find P(|Mn−1|>δ}) . (Type delta for δ .)
P(|Mn−1|>δ})=
unanswered
Does the sequence (Mn)n≥1 converge in probability to a constant? If yes, enter the value of the constant limit; if no, enter DNE.
(Mn)n≥1⟶P
unanswered
Find the CDF FMn(x) for 0≤x≤1 .
FMn(x)=P(Mn≤x)=
unanswered
Does (Mn)n≥1 converge in distribution?
Yes
No
unanswered
STANDARD NOTATION
Standard Notation
×
Guidelines Example Entries Common mistakes
Symbols These are case sensitive.
Use the correct case as specified in the problem. n and N are different. Do NOT enter x for X
Parentheses Match each open parenthesis with a close parenthesis.
Elementary arithmetic operations Use the symbols +,-,*,/ for addition, subtraction, multiplication, and division, respectively. Enter 1+b*c-d/e for 1+bc−d/e
For multiplication, use * explicitly.
Although the ''pretty'' display underneath your answer looks correct if you do not include * s, your answer will be marked incorrect! Enter b*c for bc in the example above
Enter 2*n*(n+1) for 2n(n+1) Do NOT enter bc for bc
Do NOT enter 2n(n+1) for 2n(n+1)
Exponents Use the symbol ^ to denote exponentiation. Enter 2^n for 2n
Enter x^(n+1) for xn+1
Square root use the string of letters sqrt, followed by enclosing what is under the square root in parentheses. Enter sqrt(-1) for −1−−−√
Mathematical
constants Use the symbol e for the base of the natural logarithm, e .
Use the string of letters pi for π . Enter e^(i*(pi))+1 for eiπ+1
Order of operations 1) parentheses
2) exponents and roots
3) multiplication and division
4) addition and subtraction
When in doubt, use additional parentheses to remove possible ambiguitites. Enter a/b*c for ab⋅c
Enter a/(b*c) for abc
Enter (1/sqrt(2*(pi)))*e^(-(x^2)/2) for 12π−−√e−x22
Do NOT enter a/b*c for abc
Natural logarithm Although in lectures and solved problems we will sometimes use the notation ''log'' (instead of ''ln''), you should use the string of letters ln, followed by the argument enclosed in parentheses. Enter ln(2*x) for ln(2x) Do NOT enter log(2*x) for ln(2x)
Trigonometric
functions Use the usual 3-letter symbols to denote the standard trigonometric functions Enter sin(x) for sin(x) Do NOT enter sin x for sin(x)
Greek letters Use the Latin-character name to denote each Greek letter Enter lambda*e^(-lambda*t) for λe−λt
Enter mu*alpha*theta for muαθ
Factorials, permutations, combinations Enter fact(k) or factorial(k) for k!
Do NOT use ! in your answers as it will not be evaluated correctly!
Close Guide
SubmitYou have used 0 of 3 attemptsSome problems have options such as save, reset, hints, or show answer. These options follow the Submit button.
Save Save Your Answer
Expectations and convergence in probability
3 points possible (graded)
Let (Tn)n≥1 be a sequence of r.v.s such that for each n , Tn takes only two possible values 0 and 2n with the following probabilities:
P(Tn=0) = 1−1n
P(Tn=2n) = 1n.
Does the sequence (Tn)n≥1 converge in probability to a constant? If so, enter the limiting value; if not, enter DNE.
Tn⟶P
unanswered
Compute E[Tn] in terms of n .
E[Tn]=
unanswered
Does the sequence of expectations E[Tn] converge? If so, enter the limiting value; if not, enter DNE.
limn→∞E[Tn]=
unanswered
Probability review: the (Strong) Law of Large Numbers
1 point possible (graded)
A digital signal receiver decodes bits of incoming signal as 0s or 1 and makes an error in decoding a bit with probability 10−4.
Assuming decoding success is independent for different bits, as the receiver receives more and more signals, what is the fraction of erroneously decoded bits?
Fraction of errors:
Well, that's a tough question. But let me tell you a joke to lighten the mood:
Why don't scientists trust atoms?
Because they make up everything!
The fraction of errors in decoding a bit is equal to the probability of error, which is 10^(-4).
To find the smallest number N such that P(|Tn−5|>δ)=0 whenever n>N, we need to find the value of n at which the probability becomes 0.
Let's start by finding the probability of the event {|Tn - 5| > δ}. Since Tn follows a uniform distribution, we can find the probability by calculating the area outside the interval (5 - δ, 5 + δ).
Since the distribution is symmetric around 5, we only need to calculate the probability for one side of the interval. The probability of the event {|Tn - 5| > δ} is equal to the probability of the event {Tn < 5 - δ or Tn > 5 + δ}.
Using the properties of the uniform distribution, we can calculate the probability as follows:
P(Tn < 5 - δ or Tn > 5 + δ) = P(Tn < 5 - δ) + P(Tn > 5 + δ)
Since Tn is uniformly distributed between 5 - 12n and 5 + 12n, we can substitute the values into the probability calculation:
P(Tn < 5 - δ) = P(Tn < 5 - δ) = (5 - δ - (5 - 12n)) / (5 + 12n - (5 - 12n))
Simplifying the expression:
P(Tn < 5 - δ) = (12n - δ) / (24n) = (δ / (24n)) + 1/2
P(Tn > 5 + δ) can be calculated in a similar way.
Now, to find the value of n at which the probability becomes 0, we set the probability expression equal to 0:
(δ / (24n)) + 1/2 = 0
Solving for n:
δ / (24n) = -1/2
δ = -12n
By rearranging the equation:
n = -(δ / 12)
Thus, the smallest number N (in terms of δ) such that P(|Tn−5|>δ)=0 whenever n>N is N = -(δ / 12).
Now let's move to the next question.