Question

1. Let ?1 and ?2 be two independent random variables with normal distribution with expectation 0 and variance 1.

(1) Find the covariance between ?1 + ?2 and ?1 − ?2.

(2) Find the probability that ?2 1 + ?2 2 ≤ 2. (3) Find the expectation of ?2 1 + ?2^2 .

2. Estimate the approximated value of (︂ 10000 5100 )︂ = 10000! 5100!4900! by central limit theorem.

Answer #1

2. Estimate the approximated value of (︂ 10000 5100 )︂ = 10000!
5100!4900! by central limit theorem.

Let ? and ? be two independent random variables with uniform
distribution. ?(? = 0|? = ?, ? = ?) = 1 − ?, ?(? = 1|? = ?, ? = ?)
= ?(1 − ?) and ?(? = 2|? = ?, ? = ?) = ??.
1. Find the conditional joint p.d.f. (the posterior)
??,?|?=?.
2.Write down the conditional expectation ?[?|? = ?] and ?[?|? =
?] as functions of ?.

Let ? and ? be two independent random variables with uniform
distribution. ?(? = 0|? = ?, ? = ?) = 1 − ?, ?(? = 1|? = ?, ? = ?)
= ?(1 − ?) and ?(? = 2|? = ?, ? = ?) = ??.
1.Write down the conditional expectation ?[?|? = ?] and ?[?|? =
?] as functions of ?.

3. Let ?1, ?2, ?3 be 3 independent random variables with
standard normal distribution. Find the conditional probability

Let U and V be two independent standard normal random variables,
and let X = |U| and Y = |V|.
Let R = Y/X and D = Y-X.
(1) Find the joint density of (X,R) and that of (X,D).
(2) Find the conditional density of X given R and of X given
D.
(3) Find the expectation of X given R and of X given D.
(4) Find, in particular, the expectation of X given R = 1 and of...

Let {?1,?2, … , ?? } be ? independent random draws from any
given distribution with finite expected value ? and variance ? 2
> 0. Let ?̅ = 1 ? ∑ ?? ? ?=1 denote the average draw, which in
turn is a random variable with its own distribution. This question
works through successive proofs to derive the expected value and
variance of this distribution, culminating in a proof of the Law of
Large Numbers.
a. Show that ?(??...

Let X and Y be random variables, P(X = −1) = P(X = 0) = P(X = 1)
= 1/3 and Y take the value 1 if X = 0 and 0 otherwise. Find the
covariance and check if random variables are independent.
How to check if they are independent since it does not mean that
if the covariance is zero then the variables must be
independent.

Suppose ?1 has normal distribution with expectation 0 and
variance ? 2 , ?2 has normal distribution with expectation 0 and
variance 2? 2 . ?0 : ? 2 = ?, ?1 : ? 2 > ?, where ? > 0.
Recall that the one sided ? 2 test for the null hypothesis ? ∼ ? 2
(?) is ? ≥ ? −1 ?2(?) (1 − ?), here ? is the c.d.f. of ? 2 (?), and
? is the...

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

Let X1, X2 be two normal random variables each with population
mean µ and population variance σ2. Let σ12 denote the covariance
between X1 and X2 and let ¯ X denote the sample mean of X1 and X2.
(a) List the condition that needs to be satisﬁed in order for ¯ X
to be an unbiased estimate of µ. (b) [3] As carefully as you can,
without skipping steps, show that both X1 and ¯ X are unbiased
estimators of...

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