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. (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).
Let Y denote a random variable that has a Poisson distribution
with mean λ = 3. (Round your answers to three decimal places.)
(a) Find P(Y = 6)
(b) Find P(Y ≥ 6)
(c) Find P(Y < 6)
(d) Find P(Y ≥ 6|Y ≥ 3).
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/ 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.
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.
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. (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
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 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)?