Question

Normal Distribution

Calculate the entropy of a multidimensional Gaussian p(x) = N(µ,
Σ)

Answer #1

THE NECESSARY WORKOUT HAS BEEN SHOWN ABOVE. IN CASE OF DOUBT, DO COMMENT BELOW. AND PLEASE LIKE.

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!

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 the random variable X follow a normal distribution with µ =
18 and σ = 4. The probability is 0.99 that X is in the symmetric
interval about the mean between two numbers, L and U (L is the
smaller of the two numbers and U is the larger of the two numbers).
Calculate L.

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...

Let the random variable X follow a normal distribution
with µ = 22 and σ = 4. The probability is 0.90
that Xis in the symmetric interval about the mean between
two numbers, L and U (L is the smaller of the two numbers and U is
the larger of the two numbers). Calculate U.

2. Let X be a Normal random variable with µ = 11 and σ 2 = 49.
You may refer to the tables at the end of our textbook.
(a) Calculate P(X2 > 100).
(b) Calculate the hazard rate function at 18, λ(18) and at 25,
λ(25).

Using the normal distribution, calculate the following
probabilities:
a) P(X≤16|n=50, p=0.70)
b) P(10≤X≤16|n=50, p=0.50)

Let two independent random vectors x and z have Gaussian
distributions: p(x) = N(x|µx,Σx), and p(z) = N(z|µz,Σz). Now
consider y = x + z. Use the results for Gaussian linear system to
ﬁnd the distribution p(y) for y. Hint. Consider p(x) and p(y|x).
Please prove for it rather than directly giving the result.

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...

A distribution with a mean of µ = 73 and a standard deviation of
σ = 8 is being transformed into a standardized distribution of µ =
100 and σ = 16. Find the new, standardized score for each of the
following values from the original population (plot each point on a
graph):
a. X = 80 b. X = 70
c. X = 65 d. X = 87

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 4 minutes ago

asked 7 minutes ago

asked 9 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 17 minutes ago

asked 22 minutes ago

asked 25 minutes ago

asked 44 minutes ago

asked 44 minutes ago