Question

Show that if X ∈ N(µ, σ2 ), then E(X) = µ, and V ar(X) = σ 2

Answer #1

Suppose X1, · · · , Xn from a normal distribution N(µ, σ2 )
where µ is unknown but σ is known. Consider the following
hypothesis testing problem:
H0 : µ = µ0 vs. Ha : µ > µ0
Prove that the decision rule is that we reject H0 if
X¯ − µ0 σ/√ n > Z(1 − α),
where α is the significant level, and show that this is
equivalent to rejecting H0 if µ0 is less than the...

Suppose that 10% of the probability for a certain distribution
that is N(µ, σ2 ) is below 60 and that 5% is above 90. What are the
values of
(a) µ?
(b) σ?

If X∼Binom(n,p), E[X] = np. Calculate V[X] = np(1−p) by:
a) First show E[X(X−1)] + E[X] − (E[X])^2 = V[X] (Hint: Use
propertie of E[·] and V[·]).
b) Show E[X(X−1)] = n(n−1)p^2
c) Use E[X] = np, a) and b) to discuss, V[X] = np(1−p).

Let X have the normal distribution N(µ; σ2) and let Y = eX
(a)Find the range of Y and the pdf g(y) of Y
(b)Find the third moment of Y E[Y3]
(c) In the next four subquestions, we assume that µ = 0 and σ =
1. Sketch the graph of the pdf of Y for 0<y<=5 (use Maple to
generate the graph and copy it the best you can in the answer
box)
(d)What is the mean of Y...

If V ar(X)=2, then V ar(8X+13)=?

Suppose that we use x to estimate the mean of x, when E[x] = µ,
Var[x] = σ 2 . Further suppose that both µ and σ 2 have finite
values. As the sample size n gets larger, the variance of x gets
closer to _______ ?

If X ∼ N(µ, σ) then Y = e^X has a log(Y) that has a Normal
distribution.
1. without calculating, explain if E(Y) is greater than, less
than, or equal to e^u.
2. Calculate E(Y)
3. Find the pdf of Y and sketch a plot of it

Let X ∼ N (µ, σ^2). Prove that P (|X − µ| > kσ) does not
depend on µ or σ.
Please write your answer as clearly as you can, appreciate
it!

R Simulation:
For n = 10, simulate a random sample of size n from N(µ,σ2),
where µ = 1 and σ2 = 2; compute the sample mean. Repeat the above
simulation 500 times, plot the histogram of the 500 sample means
(does it mean that I can just use hist() method instead of plot()
method). Now repeat the 500 simulations for n = 1,000. Compare
these two sets of results with diﬀerent sample sizes, and discuss
it in the context...

Let X and Y be independent and normally distributed random
variables with waiting values E (X) = 3, E (Y) = 4 and variances V
(X) = 2 and V (Y) = 3.
a) Determine the expected value and variance for 2X-Y
Waiting value µ = Variance σ2 = σ 2 =
b) Determine the expected value and variance for ln (1 + X
2)
c) Determine the expected value and variance for X / Y

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