Download 1. Let f(x) be an odd function. Show that ∫ ∞ f(x)dx = 0 - Math-UMN

1. Let f (x) be an odd function. Show that
Z ∞
f (x)dx = 0
−∞
2. Show that the product of an odd function and an even function is an odd
function.
3 X−µ
, where µ and σ
3. The skew of a random variable X is given by E
σ
are the mean and variance, respectively. Show that the skew of a random
variable with density symmetric about µ is 0.
4. Let X have density Φµ,σ2 . Prove the following equalities
1
2
E eX
= eµ+ 2 σ
E e2X
= e2µ+2σ
2
What is V ar(eX )?
5. Let Y be a lognormally distributed random variable with Y = eX , and
X having mean µ and standard deviation σ. Find the probability that
eµ−kσ ≤ Y ≤ eµ+kσ for k = 1, . . . , 6 and µ = 0.0005 and σ = 0.018.
6. Let {Xi }M
i=1 be iid random variables. Let the mean, µ, be 0.0005 and the
standard deviation be 0.018. If
MN =
find N such that P |MN − µ| >
µ
10
N
1 X
Xi
N i=1
≤ .01.
7. Let the joint density for a bivariate normal distribution be given by
2
1
x − 2ρxy + y 2
p
f (x, y) =
exp −
.
2(1 − ρ2 )
2π (1 − ρ2 )
Show that Corr(X 2 , Y 2 ) = ρ2 . [2.5 in SMMF]
1