Let Yi equal a normal distribution with mean β + βxi and
variance σ2 = 1...
Let Yi equal a normal distribution with mean β + βxi and
variance σ2 = 1 for i = 1; 2; :::; n. Further, suppose that the Yi
are independent and that the xi are known values. Find the MLE of β
0 and β1.
Let {Zt} be independent normal random variables with mean 0 and
variance σ2. Let a, b, c be constants. Which of the
following processes are stationary? Evaluate mean and
autocovariance function.
(a) Xt = Ztcos(at) +
Zt−1sin(bt)
(b) Xt =a+bZt + cZt−2
(c) Xt = ZtZt−1
1. Let X have a normal distribution with parameters μ = 50 and
σ2 =
144. Find the probability that X produces a value between 44 and
62. Use the
normal table A7 (be sure to show your work).
2. Let X ~ Exponential( λ ), for some fixed constant λ > 0.
That is,
fX(x) = λ e-λx = λ exp( -λx ), x > 0, ( fX(x) = 0
otherwise)
(a) Create a transformed random variable Y =...
Let X have Normal distribution with mean 45 and variance 81. If
a random sample of size 25 is taken, which of the following is the
probability that the sample average is between 41.40 and 45.63?
let the random variable x follow a normal distribution with μ =
50 and σ2 = 64.
a. find the probability that x is greater than 60.
b. find the probability that x is greater than 35 and less than
62
. c. find the probability that x is less than 55.
d. the probability is 0.2 that x is greater than what
number?
e. the probability is 0.05 that x is in the symmetric interval
about the mean between...
Let X be a random variable which follows normal distribution
with mean 12 and variance 0.25. Then find the following
probabilities
(a) P( X ≤ 15 )
(b) P( X ≤ 17.5 )
(c) P( |X-15| ≤ 3 )
Assume a normal population with known variance σ2, a random
sample (n< 30) is selected. Let x¯,s represent the sample mean
and sample deviation. (1)(2pts) write down the formula: 98%
one-sided confidence interval with upper bound for the
population mean. (2)(6pts) show how to derive the confidence
interval formula in (1).
Let X be a random variable with mean μ and variance σ2.
Given two independent random samples of sizes n1
= 9 and n2
= 7, with sample means X1-bar
and X2-bar,
if
X-bar = k X1-bar
+ (1 – k) X2-bar,
0 < k < 1, is an unbiased estimator for μ. If X1-bar
and X2-bar
are independent, find the value of k that minimizes the standard
error of X-bar.
Mean and Variance of:
continuous uniform distribution,
normal distribution,
normal approximation for binomial and Poisson
distributions,
exponential distribution.
and Continuous random variables Cumulative Distribution
Function.
In your own words summarize the above in three to five long
sentences