# Probability

Problem 1

Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X2]=E[Y2]=E[Z2]=5.

Find cov(XY,XZ).

cov(XY,XZ)= ?

Problem 2.

Let X be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from X). In case of Heads, we let Y=X. In case of Tails, we let Y=−X.

yes
no
not enough information to determine

2) Compute Cov(X,Y).
Cov(X,Y)= ?

Are X and Y independent?
yes
no
not enough information to determine

Problem 3.

2.0 points possible (graded, results hidden)
Find P(X+Y≤0).

P(X+Y≤0)= ?

1. 👍 1
2. 👎 0
3. 👁 580
1. Use Cov(XY,XZ)=E( XY XZ)E(XY)E(XZ) =E(X2)E(Y)E(Z)E(X)E(Y)E(X)E(Z). =5

1. 👍 3
2. 👎 9
2. you post a complicated statistical formula, yet you have trouble with a simple quadratic equation?

What's wrong with this picture?

1. 👍 2
2. 👎 2
3. E[XYXZ]-E[XY]*E[XZ]= E[X^2]*E[Y]*E[Z]-E[X]*E[Y]*E[X]*E[Z]

Because X is independent of Y and Z, X^2 is also independent of Y and Z

1. 👍 2
2. 👎 0
4. is the Y is normal

1. 👍 1
2. 👎 0
5. Y is normal, did anyone found the Cov(X,Y)? Or the probability?

I found that Cov=1 and P(X+Y)=3/4

But I am not sure...

1. 👍 0
2. 👎 0
6. Can you write steps to your solution?

1. 👍 0
2. 👎 0
7. X is a standard normal, E[X]=0 and Var(X)=1

Cov(X,Y) = E[XY] - E[X]E[Y]. Since Y=|X|, Cov(X,Y) = E[X^2] - E[X].E[X] = Var(X) --> E[X^2] = 1 --> Cov(X,Y) = 1. They are not independent because is different than 0.

The probability question I found it intuitively,
for all values of X=-Y, the results are 0, so 1/2(fair coin toss)*1

for X=Y, half of the values are minor or equal to 0, so 1/2(fair coin toss)*1/2

P(X+Y)= (1/2*1) + (1/2*1/2) = 3/4

Does that makes sense?

1. 👍 1
2. 👎 0
8. I think that it does.

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Probability

For each of the following statements, determine whether it is true (meaning, always true) or false (meaning, not always true). Here, we assume all random variables are discrete, and that all expectations are well-defined and

2. ### probability

Consider n independent rolls of a k-sided fair die with k≥2: the sides of the die are labelled 1,2,…,k and each side has probability 1/k of facing up after a roll. Let the random variable Xi denote the number of rolls that

3. ### Statistics

Suppose a basketball player is an excellent free throw shooter and makes 90% of his free throws (i.e., he has a 90% chance of making a single free throw). Assume that free throw shots are independent of one another. Suppose this

4. ### Probability

Suppose that we have a box that contains two coins: A fair coin: P(H)=P(T)=0.5 . A two-headed coin: P(H)=1 . A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice. Conditioned

1. ### Probability

1. Suppose three random variables X , Y , Z have a joint distribution PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). Then, are X and Y independent given Z? 2.Suppose random variables X and Y are independent given Z , then the

2. ### Probability

Suppose that we have a box that contains two coins: A fair coin: P(H)=P(T)=0.5 . A two-headed coin: P(H)=1 . A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice. Conditioned

3. ### Probability

1.Let 𝑋 and 𝑌 be two binomial random variables: a.If 𝑋 and 𝑌 are independent, then 𝑋+𝑌 is also a binomial random variable b.If 𝑋 and 𝑌 have the same parameters, 𝑛 and 𝑝 , then 𝑋+𝑌 is a binomial

4. ### math, probability

Let X and Y be independent random variables, uniformly distributed on [0,1] . Let U=min{X,Y} and V=max{X,Y} . Let a=E[UV] and b=E[V] 1. Find a 2. Find b 3. Find Cov(U,V) . You can give either a numerical answer or a symbolic

1. ### Probability

Let X1 , X2 , X3 be i.i.d. Binomial random variables with parameters n=2 and p=1/2 . Define two new random variables Y1 =X1−X3, Y2 =X2−X3. We further introduce indicator random variables Zi∈{0,1} with Zi=1 if and only if

2. ### statistics

Suppose that Θ , X1 , and X2 have zero means. Furthermore, Var(X1)=Var(X2)=Var(Θ)=4, and Cov(Θ,X1)=Cov(Θ,X2)=Cov(X1,X2)=1. The LLMS estimator of Θ based on X1 and X2 is of the form Θˆ=a1X1+a2X2+b . Find the coefficients a1

3. ### Probability

For each of the following statements, state whether it is true (meaning, always true) or false (meaning, not always true): 1. Let X and Y be two binomial random variables. (a) If X and Y are independent, then X+Y is also a

4. ### Math

Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X2]=E[Y2]=E[Z2]=5. Find cov(XY,XZ).