Question

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

Answer #1

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...

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

If X is a normal random variable that has a mean of µ = 20 and a
standard deviation σ = 2, (a) the standardized value of X=16 is
_________. (b) What is the probability that X is less than or equal
to 16? __________ (c) What is the probability that X is greater
than 16? __________ (d) What is the probability that X is equal to
16?________

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.

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.

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

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 =
19 and σ2 = 8. Find the probability that X is greater than 11 and
less than 15.

Let the random variable X follow a normal distribution with µ =
18 and σ2 = 11. Find the probability that X is greater than 10 and
less than 17.

1. Suppose that X has an Exponential distribution with rate
parameter λ = 1/4. Also suppose that given X = x, Y has a
Uniform(x, x + 1) distribution.
(a) Sketch a plot representing the joint pdf of (X, Y ). Your
plot does not have to be exact, but it should clearly display the
main features. Be sure to label your axes.
(b) Find E(Y ).
(c) Find Var(Y ).
(d) What is the marginal pdf of Y ?

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