I need help with this problem. Please explain in detail.
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I need help with this problem. Please explain in detail.
7. (based on 3.3.9) Let X...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Let A > 0 be fixed and for each n - 1,2,3.., let Xn be a...
Let A > 0 be fixed and for each n - 1,2,3.., let Xn be a Binomial Random variable with parameters n, and pn -^. (i.e The number of trials is n and thıe success probability is pn --) (a) Write the moment-generating-function, Mx (t of X,. (You do not have to 72 derive it from scratch. You may use the general formula for the mgf of a binomial variable as provided in the appendix of the text). (b) Show...
Please help a) How many parameters does a mixture of m Gaussians have? b) Let x1,...
Please help
a) How many parameters does a mixture of m Gaussians
have?
b) Let x1, . . . , xn be n observations drawn from a mixture
of m Gaussians. Write down the log-likelihood function. （Hint: it
should involve two summations.）
c) Let 1 ≤ k ≤ m. Show that the maximum likelihood estimator
for µk is given b
and d)
A mixture of m univariate Gaussians has the PDF TIL where each P3 > 0 and Σ-1 pi-|...
need the solution for this question.tq Let X,,. X, be a random sample from a Poisson (a) (a). 2. distribution. Find...
need the solution for this question.tq
Let X,,. X, be a random sample from a Poisson (a) (a). 2. distribution. Find the sufficient statistic for A. (25 marks) Let X,X,X, be a random sample from a gamma (k, B) (b). P.1 distribution with k is fixed. DefineX X, n피 based upon unbiased ness, consistency Evaluate (0). and efficiency is a minimum variance unbiased estimator for B Show that (ii). (75 marks) (2)3
Let X,,. X, be a random sample from...
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a)...
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
(f) (g) (e) and (f) the last 4 questions are what i need help with Problem(12)...
(f)
(g) (e) and (f) the last 4 questions are what i need help
with
Problem(12) Let X.X... X, be i.i.d. r.v's from Normal, a'). (a) (1 pt) What is the distribution of Mer? (b) (1 pt) What is the distribution of (S.)? (c) (1 pt) What is the distribution of (NL) (d) (7 pts) Show that (1) (e) (1 pt) What is the distribution of (1) S? (1) (3 pts) Simplify (B) (1 pt) What is the distribution of...
(b) and (c) are what i need help with Problem(7) Let z Student's t-distribution. The density...
(b)
and (c) are what i need help with
Problem(7) Let z Student's t-distribution. The density function of T is Normal(0, 1) and Y ~ xã, then the new r.v. T = Jun has the r[(y + 1)/2) -(4+1)/2 fr(t) = + (7/2) (a) (3 points) Describe the similarity/difference between T and Z. (b) (6 points) Let to be a particular value of t. Use t-distribution table to find to values such that the following statements are true. (Given that...
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for...
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e
Let X have a Weibull distribution with parameters a and B, So Е(X) 3 В ....
Let X have a Weibull distribution with parameters a and B, So Е(X) 3 В . Г(1 VX) %3D в2{г(1 1/a) + [r(1/a)2 2/a) (a) Based on a random sample X1, . . . , Xn, write equations for the method of moments estimators of B and a. Show that, once the estimate of a has been obtained, the estimate of B can be found from a table of the gamma function and that the estimate of a is the...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Let A > 0 be fixed and for each n - 1,2,3.., let Xn be a Binomial Random variable with parameters n, and pn -^. (i.e The number of trials is n and thıe success probability is pn --) (a) Write the moment-generating-function, Mx (t of X,. (You do not have to 72 derive it from scratch. You may use the general formula for the mgf of a binomial variable as provided in the appendix of the text). (b) Show...
Please help
a) How many parameters does a mixture of m Gaussians
have?
b) Let x1, . . . , xn be n observations drawn from a mixture
of m Gaussians. Write down the log-likelihood function. （Hint: it
should involve two summations.）
c) Let 1 ≤ k ≤ m. Show that the maximum likelihood estimator
for µk is given b
and d)
A mixture of m univariate Gaussians has the PDF TIL where each P3 > 0 and Σ-1 pi-|...
need the solution for this question.tq
Let X,,. X, be a random sample from a Poisson (a) (a). 2. distribution. Find the sufficient statistic for A. (25 marks) Let X,X,X, be a random sample from a gamma (k, B) (b). P.1 distribution with k is fixed. DefineX X, n피 based upon unbiased ness, consistency Evaluate (0). and efficiency is a minimum variance unbiased estimator for B Show that (ii). (75 marks) (2)3
Let X,,. X, be a random sample from...
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
(f)
(g) (e) and (f) the last 4 questions are what i need help
with
Problem(12) Let X.X... X, be i.i.d. r.v's from Normal, a'). (a) (1 pt) What is the distribution of Mer? (b) (1 pt) What is the distribution of (S.)? (c) (1 pt) What is the distribution of (NL) (d) (7 pts) Show that (1) (e) (1 pt) What is the distribution of (1) S? (1) (3 pts) Simplify (B) (1 pt) What is the distribution of...
(b)
and (c) are what i need help with
Problem(7) Let z Student's t-distribution. The density function of T is Normal(0, 1) and Y ~ xã, then the new r.v. T = Jun has the r[(y + 1)/2) -(4+1)/2 fr(t) = + (7/2) (a) (3 points) Describe the similarity/difference between T and Z. (b) (6 points) Let to be a particular value of t. Use t-distribution table to find to values such that the following statements are true. (Given that...
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e
Let X have a Weibull distribution with parameters a and B, So Е(X) 3 В . Г(1 VX) %3D в2{г(1 1/a) + [r(1/a)2 2/a) (a) Based on a random sample X1, . . . , Xn, write equations for the method of moments estimators of B and a. Show that, once the estimate of a has been obtained, the estimate of B can be found from a table of the gamma function and that the estimate of a is the...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
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