Homework Answers
Add Answer to:
omly the converse of part b.
Suppose (xn) is decreasing and nonnegative. Let Sn:= x1 +...
Exercise 5.22. Let (Xn)nel be a sequence of i.i.d. Poisson(a) RVs. Let Sn-X1++Xn (i) Let Zn-(Sn-nA)/Vm....
Exercise 5.22. Let (Xn)nel be a sequence of i.i.d. Poisson(a) RVs. Let Sn-X1++Xn (i) Let Zn-(Sn-nA)/Vm. Show that as n-, oo, Zn converges to the standard normal RV Z ~ N(0,1) in distribution (ii) Conclude that if Yn~Poisson(nX), then ii) Fromii) deduce that we have the following approximation which becomes more accurate as noo.
Let X1, X2,...be a sequence of random variables. Suppose that Xn?a in probability for some a...
Let X1, X2,...be a
sequence of random variables. Suppose that Xn?a in probability for
some a ? R. Show that (Xn) is Cauchy convergent in probability,
that is, show that for all
> 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true?
(Prove if “yes”, find a counterexample if “no”)
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An...
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information co...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let X0, X1, X2,... be a branching process (as defined in class), i.e. Xn gives then number of ind...
Let X0, X1, X2,... be a
branching process (as defined in class), i.e. Xn gives
then number of individuals in the nth generation. Suppose that the
mean number of offspring per individual is μ. Show that
Mn = μ-nXn is a martingale with
respect to X0, X1, X2,...
Let Xo, X1, X2,... be a branching process (as defined in class), i.e., Xn gives then number of individuals in the nth generation. Suppose that the mean number of offspring per individual...
Theorem 5.10: Please answer both parts Problem 7. Let Sn = X1 + ... + Xn,...
Theorem 5.10:
Please answer both parts
Problem 7. Let Sn = X1 + ... + Xn, where Xį are independent with EX; = 0 and Var(X;) = 02. By example 5.3, p. 205, S2 – no2 is a martingale. Let Ta = min{n : Snl > a}. (a) Use Theorem 5.10, to show that E[Ta] > . (b) Show that, for simple random walk, o2 = 1, we have equality. Theorem 5.10. If Mn is a supermartingale with respect to...
Let X, and Sn be the sample mean and the sample variance of {Xi,.X. Let Xn+1...
Let X, and Sn be the sample mean and the sample variance of {Xi,.X. Let Xn+1 and S7+1 be the sample mean and the sample variance of {X1,..., Xn, X y. Which of the following hold for sample means, for sample variances? 72 m+1 m+1
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
Let X0,X1,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following...
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
Exercise 5.22. Let (Xn)nel be a sequence of i.i.d. Poisson(a) RVs. Let Sn-X1++Xn (i) Let Zn-(Sn-nA)/Vm. Show that as n-, oo, Zn converges to the standard normal RV Z ~ N(0,1) in distribution (ii) Conclude that if Yn~Poisson(nX), then ii) Fromii) deduce that we have the following approximation which becomes more accurate as noo.
Let X1, X2,...be a
sequence of random variables. Suppose that Xn?a in probability for
some a ? R. Show that (Xn) is Cauchy convergent in probability,
that is, show that for all
> 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true?
(Prove if “yes”, find a counterexample if “no”)
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let X0, X1, X2,... be a
branching process (as defined in class), i.e. Xn gives
then number of individuals in the nth generation. Suppose that the
mean number of offspring per individual is μ. Show that
Mn = μ-nXn is a martingale with
respect to X0, X1, X2,...
Let Xo, X1, X2,... be a branching process (as defined in class), i.e., Xn gives then number of individuals in the nth generation. Suppose that the mean number of offspring per individual...
Theorem 5.10:
Please answer both parts
Problem 7. Let Sn = X1 + ... + Xn, where Xį are independent with EX; = 0 and Var(X;) = 02. By example 5.3, p. 205, S2 – no2 is a martingale. Let Ta = min{n : Snl > a}. (a) Use Theorem 5.10, to show that E[Ta] > . (b) Show that, for simple random walk, o2 = 1, we have equality. Theorem 5.10. If Mn is a supermartingale with respect to...
Let X, and Sn be the sample mean and the sample variance of {Xi,.X. Let Xn+1 and S7+1 be the sample mean and the sample variance of {X1,..., Xn, X y. Which of the following hold for sample means, for sample variances? 72 m+1 m+1
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
Most questions answered within 3 hours.
-
9. (9 pts) An investor with a stock portfolio worth several
hundred thousand dollars sued his...
asked 14 minutes ago -
Which statement is true?
a. Unless directed otherwise, the computer automatically
executes C. statements one before...
asked 35 minutes ago -
When a new company is considering Speed to Market for a product
launch, which would provide...
asked 1 hour ago -
bacteria and their metabolites could either promote or protect from
cancer. Explain one of these cases...
asked 1 hour ago -
Use the sample data and confidence level given below to complete
parts (a) through (d).
A...
asked 2 hours ago -
During laser vision correction, a brief burst of 193 nm
ultraviolet light is projected onto the...
asked 2 hours ago -
1 ) How well does the initial speed, v0,calculated from Equation
4 agree with the value...
asked 2 hours ago -
1. What is the p-value for a right tail test for a single
proportion with a...
asked 3 hours ago -
A technician compares repair costs for two types of microwave
ovens (type I and type II)....
asked 2 hours ago -
A certain statistical test is designed so that it fails 2% of
the time. Suppose that...
asked 2 hours ago -
For the reaction: H2S(g) + 2H2O(l)3H2(g) + SO2(g) H° = 295 kJ
and S° = 295...
asked 3 hours ago -
The first-order reaction C4H8→2C2H4 has the reaction constant k
of 7.60×10−3 s−1. If the initial concentration...
asked 3 hours ago