Question

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

Answer #1

**Solution:**

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.

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 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...

1)Let X1, ..., Xn be independent standard normal random
variables, we know that X2 1 + ... + X2 n follows the chi-squared
distribution of n degrees of freedom. Find the third moment of the
the chi-squared distribution of 2 degrees of freedom.
2) Suppose that, on average, 1 person in 1000 makes a numerical
error in preparing his or her income tax return. If 10,000 returns
are selected at random and examined, find the probability that 6 or
7...

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 ?.

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, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

Let X and Y be two independent random variables with ??=3, ??=2,
??=6, and ??=1.
Find ?(5?−3?+2)−?(8?−3?+7).
Your answer should be a whole number.

Let X and Y be two random variables which follow standard normal
distribution. Let J = X − Y . Find the distribution function of J.
Also find E[J] and Var[J].

Let z be a random variable with a standard normal distribution.
Find the indicated probability. (Round your answer to four decimal
places.) P(z ? ?0.25)
Let z be a random variable with a standard normal distribution.
Find the indicated probability. (Round your answer to four decimal
places.) P(z ? 1.24)
Let z be a random variable with a standard normal distribution.
Find the indicated probability. (Enter your answer to four decimal
places.) P(?2.20 ? z ? 1.08)

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