Question

Consider the sample x_{1}, x_{2}, ...,
x_{n} with sample mean x̅ and sample standard deviation s
.Let Z_{i} = (x_{i} - x̅ )/s, i = 1,2, ..., n. What
are the values of the sample mean and sample standard deviation of
z_{i} ? Explain the answers with equations.

Answer #1

given that

Consider the sample x1, x2, ..., xn with sample mean x̅ and sample standard deviation s .Let Zi = (xi - x̅ )/s, i = 1,2, ..., n.

, i = 1,2,...,n.

sample mean

sample standard deviation =

sample standard deviation

Let X1, X2,...,Xn represent n random draws from a population
with standard deviation σ and variance σ^2 , so that V ar[X1] = V
ar[X2] = ... = V ar[Xn] = σ^ 2 . Define the sample average taken
from a sample of size n as follows: X¯ n ≡ (X1 + X2 + ... + Xn)/ n
.
a) Derive an expression for the standard deviation of X¯ n.
[Hint: Your answer should depend only on σ and n]...

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

Suppose n numbers X1, X2, . . . , Xn are chosen from a uniform
distribution on [0, 10]. We say that there is an increase at i if
Xi < Xi+1. Let I be the number of increases. Find E[I].

Let X1, X2, . . ., Xn be independent, but not identically
distributed, samples. All these Xi ’s are assumed to be normally
distributed with
Xi ∼ N(θci , σ^2 ), i = 1, 2, . . ., n,
where θ is an unknown parameter, σ^2 is known, and ci ’s are
some known constants (not all ci ’s are zero). We wish to estimate
θ.
(a) Write down the likelihood function, i.e., the joint density
function of (X1, ....

Let X1, X2, . . . , Xn be a random sample of size n from a
distribution with variance σ^2. Let S^2 be the sample variance.
Show that E(S^2)=σ^2.

Let X1,...,Xn be a sample drawn from a normal population with
mean μ and standard deviation σ. Find E[X ̄S2].

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Suppose that we have a random sample X1, ** , Xn drawn from a
distribution that only takes positive values. Suppose that the
sample size n is sufficiently large. Consider the new random
variable ∏ n i=1 Xi . Derive the distribution of this new random
variable and explain your reasoning mathematically

Suppose that we have a random sample X1, · · , Xn drawn from a
distribution that only takes positive values. Suppose that the
sample size n is sufficiently large. Consider the new random
variable ∏ n i=1 Xi . Derive the distribution of this new random
variable and explain your reasoning mathematically

Let Xi, i=1,...,n be independent exponential r.v. with mean
1/ui. Define Yn=min(X1,...,Xn), Zn=max(X1,...,Xn).
1. Define the CDF of Yn,Zn
2. What is E(Zn)
3. Show that the probability that Xi is the smallest one among
X1,...,Xn is equal to ui/(u1+...+un)

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