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 Yi=0 for i=1,2 .

Calculate the covariance of Y1 and Y2 .

1. cov(Y1,Y2)
2. var(Z1)
3. cov(Z1,Z2)

1. 👍
2. 👎
3. 👁
1. 1. 1/2
2. 15/64
3. Don't know how to do

1. 👍
2. 👎
2. can you please throw some light on, how is it 15/64 ? I can add for the next step

1. 👍
2. 👎
3. no. 2) 3/16

1. 👍
2. 👎
4. for 1: I am getting
Cov(Y1,Y2) = ... = E(X3^2) = var(X3)+(E(X3))^2 = 1/2 + (1)^2 = 3/2 ?
E[X3]=n.p = (2.1/2)=1, var(x3) = n.p(1-p)=(X3)=n.2.1/2.1/2 = 1/2

1. 👍
2. 👎
5. Correction: Cov(Y1,Y2) = ... = C(X3,X3) = var(X3) = 1/2

1. 👍
2. 👎
6. Why would the variance of Z1 be 15/64?

1. 👍
2. 👎
7. How much did you get?

1. 👍
2. 👎
8. I got 1/4. I found this from google search:
"Your Z=X−Y will not be a "shifted binomial" unless p=1/2, or the trivial cases where at least one of n and m is zero. For the case p=1/2, m−Y has the same distribution as Y so X+Y and X−Y+m have the same distribution, which is indeed binomial.

In general consider the means and variances of the distributions:

X has mean np and variance np(1−p)
Y has mean mp and variance mp(1−p)
X+Y has mean (n+m)p and variance (n+m)p(1−p)
Z=X−Y has mean (n−m)p and variance (n+m)p(1−p)"

which makes probability of success of y1 =1/2. So, so, var(z1) = p(1-p)=1/4

1. 👍
2. 👎
9. So, so ...and hat happend whit term (n+m) on (n+m)p(1-p)?

1. 👍
2. 👎
10. Z is an Indicator random variable, It is not direct a substract of tow binomial distribuction

1. 👍
2. 👎
11. for 3:

to me it looks that knowing Z1 doesn't inform you for Z2 and vice versa, so they look independent variables to me -> Cov(z1,z2)= 0. (couldn't figure anything else and made up this nice explanation to comfort my self

1. 👍
2. 👎

Similar Questions

1. STATISTICS

Consider a binomial random variable where the number of trials is 12 and the probability of success on each trial is 0.25. Find the mean and standard deviation of this random variable. I have a mean of 4 and a standard deviation

2. Statistics

Let X1,X2,…,Xn be i.i.d. random variables with mean μ and variance σ2 . Denote the sample mean by X¯¯¯¯n=∑ni=1Xin . Assume that n is large enough that the central limit theorem (clt) holds. Find a random variable Z with

3. Statistics

Z1,Z2,…,Zn,… is a sequence of random variables that converge in distribution to another random variable Z ; Y1,Y2,…,Yn,… is a sequence of random variables each of which takes value in the interval (0,1) , and which

4. 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

1. statistic

Quiz has 6 questions. Each question has five possible answers, only one of each 5 answers is correct. If student randomly guesses on all six questions, what is the probability to answer 2 questions right? Tip: first, you have to

2. Statistics

1) A Motor Company has purchased steel parts from a supplier for several years and has found that 10% of the parts must be returned because they are defective. An order of 25 parts is received. What are the mean and standard

3. Math

For the discrete random variable X, the probability distribution is given by P(X=x)= kx x=1,2,3,4,5 =k(10-x) x=6,7,8,9 Find the value of the constant k E(X) I am lost , it is the bonus question in my homework on random variables

4. probability

This figure below describes the joint PDF of the random variables X and Y. These random variables take values in [0,2] and [0,1], respectively. At x=1, the value of the joint PDF is 1/2. (figure belongs to "the science of

1. probability

t the discrete random variable X be uniform on {0,1,2} and let the discrete random variable Y be uniform on {3,4}. Assume that X and Y are independent. Find the PMF of X+Y using convolution. Determine the values of the constants

2. Probability & Statistics

Exercise: Convergence in probability a) Suppose that Xn is an exponential random variable with parameter λ=n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with parameter

3. Statistics

Let X1,…,Xn be i.i.d. Poisson random variables with parameter λ>0 and denote by X¯¯¯¯n their empirical average, X¯¯¯¯n=1n∑i=1nXi. Find two sequences (an)n≥1 and (bn)n≥1 such that an(X¯¯¯¯n−bn) converges in

4. Probability and Statistics

The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6. To test this hypothesis, a random sample of 15 adults is selected. If the number of college graduates in the sample is anywhere