Stirling’s Formula, which gives approximation for factorials,
can be derived using CLT.
(a) Suppose that X1, X2, · · · , Xn is an i.i.d. sample from
Exp(1). Show that, for a standard normal
random variable Z,
(b) Show
by differencing both sides of the approximation in part a. Then set x = 0 to get Stirling’s Formula.
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Stirling’s Formula, which gives approximation for factorials,
can be derived using CLT.
(a) Suppose that X1,...
tirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1,...
tirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, , Xn is an ii.d. sample from Exp(1). Show that, for a standard normal (b) Show by differencing both sides of the approximation in part a. Then set 0 to get Stirling's Formula.
3. Suppose that X1,X2, ,Xn are i.id. N(0, σ2). Find a function of T(X)-Σǐii verges in...
3. Suppose that X1,X2, ,Xn are i.id. N(0, σ2). Find a function of T(X)-Σǐii verges in distribution to a normal distribution. State the mean and variance of your limiung normal distribution. 4. Stirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, .Xn is an ii.d. sample from Exp(1). Show that, for a standard normal PTPZ) (b) Show by differencing both sides of the approximation in part a. Then set...
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for...
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z b) Show Г(n) by differencing both sides of the approximation in part a. Then set a -0 to get Stirling's Formula. 5. Suppose that Y is an id sample from Negative Binomial (n,p). Give a normal approximation of Yn use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Ai, converges to...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1),...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the density f(x) = 1 µ e −x/µ for x > 0. (a) Let ˆµ1 = X. Show ˆµ1 is a minimum variance unbiased estimator. (b) Let ˆµ2 = (X1 +X2)/2. Show ˆµ2 is unbiased. Calculate V ar(ˆµ2). Confirm V ar(ˆµ1) < V ar(ˆµ2). Calculate the efficiency of ˆµ2 relative to ˆµ1. (c) Show X is consistent and sufficient. (d) Show ˆµ2 is not consistent...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
X1,...,Xn are IID with N(0,2). a) Determine the mean and variance for (X (subscript 1)^2) b)...
X1,...,Xn are IID with N(0,2).
a) Determine the mean and variance for (X (subscript 1)^2)
b) Show
sqrt(n)
* [ log ( 1/n ∑(from i=1 to n)
Xi2) − log(σ2 ) ] d → N(0, 2).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with...
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
tirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, , Xn is an ii.d. sample from Exp(1). Show that, for a standard normal (b) Show by differencing both sides of the approximation in part a. Then set 0 to get Stirling's Formula.
3. Suppose that X1,X2, ,Xn are i.id. N(0, σ2). Find a function of T(X)-Σǐii verges in distribution to a normal distribution. State the mean and variance of your limiung normal distribution. 4. Stirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, .Xn is an ii.d. sample from Exp(1). Show that, for a standard normal PTPZ) (b) Show by differencing both sides of the approximation in part a. Then set...
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z b) Show Г(n) by differencing both sides of the approximation in part a. Then set a -0 to get Stirling's Formula. 5. Suppose that Y is an id sample from Negative Binomial (n,p). Give a normal approximation of Yn use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Ai, converges to...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
X1,...,Xn are IID with N(0,2).
a) Determine the mean and variance for (X (subscript 1)^2)
b) Show
sqrt(n)
* [ log ( 1/n ∑(from i=1 to n)
Xi2) − log(σ2 ) ] d → N(0, 2).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
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