Question
In: Statistics and Probability
Suppose γ is a random variable with Exp(θ) distribution. Conditioning on γ, Y ∼ Poisson(γ). Provide...
Suppose γ is a random variable with Exp(θ) distribution. Conditioning on γ, Y ∼ Poisson(γ). Provide the marginal mean and variance of Y
Solutions
orchestra answered 1 year agoRelated Solutions
Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we...
Suppose X and Y are independent random variables with Exp(θ = 2)
distribution. Note that, we say X ∼ Exp(θ) if its pdf is f(x) = 1/θ
e^(−x/θ) , for x > 0 and θ > 0.
(a) What is the joint probability density function (pdf) of (X,
Y )?
(b) Use the change of variable technique (transformation
technique) to evaluate the joint pdf fW,Z (w, z) of (W, Z), where W
= X −Y and Z = Y ....
Let Y denote a random variable that has a Poisson distribution with mean λ = 6....
Let Y denote a random variable that has a Poisson
distribution with mean λ = 6. (Round your answers to three
decimal places.)
(a) Find P(Y = 9).
(b) Find P(Y ≥ 9).
(c) Find P(Y < 9).
(d) Find P(Y ≥
9|Y ≥ 6).
For a random variable X with a Cauchy distribution with θ = 0 , so that...
For a random variable X with a Cauchy distribution with θ = 0 ,
so that
f(x) =(1/ π)/( 1 + x^2) for -∞ < x < ∞
(a) Show that the expected value of the random variable X does not
exist.
(b) Show that the variance of the random variable X does not
exist.
(c) Show that a Cauchy random variable does not have finite moments
of order greater than or equal to one.
4. Suppose that the random variable Y~U(0,θ). a. Use a transformation technique to show that W=Y/...
4. Suppose that the random variable Y~U(0,θ).
a. Use a transformation technique to show that W=Y/ θ is a
pivotal quantity.
b. Use a. to find a 95% confidence interval for θ.
c. The time Clark walks into his Stats class is U(0,θ), where 0
represents 4:55pm, and time is measured in minutes after 4:55. If
he shows up at 5:59 on a given day, use c. to find a 95% confidence
interval for θ. Use two decimal places. Interpret...
The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0.
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Crameer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?
The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 or -(x/θ)^2 if you cannot read that)
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
The random variable X is distributed with pdffX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please...
The random variable X is distributed with pdffX(x, θ)
= c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 - that is
-(x/θ)^2 if your computer doesn't work)
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB)....
The demand for a replacement part is a random variable having a Poisson probability distribution with...
The demand for a replacement part is a random variable having a
Poisson probability distribution with a mean of 4.7. If there are 6
replacement parts in stock, what is the probability of a stock-out
(i.e., demand exceeding quantity on hand)?
X is a random variable following Poisson distribution. X1 is an observation (random sample point) of...
X is a random variable following Poisson distribution. X1 is an
observation (random sample point) of X.
(1.1) Please find probability distribution of X and X1. Make
sure to define related parameter properly.
(1.2) Please give the probability distribution of a random
sample with sample size of n that consists of X1, X2, ..., Xn as
its observations.
(1.3) Please give an approximate distribution of the sample mean
in question 1.2(say, called Y) when sample size is 100 with
detailed...
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