The process to generate the two RVs is as follows. We first draw T from Uniform(0,1).

Zoagliaj 2022-07-16 Answered
The process to generate the two RVs is as follows. We first draw T from U n i f o r m ( 0 , 1 ) . If T 0.5 we take X = T and draw Y from U n i f o r m ( 0 , 1 ) . Otherwise if T > 0.5 , we take Y = T and draw X from U n i f o r m ( 0 , 1 ) . Running a simulation it seems like X and Y are positively correlated, though intuitively it seems like they should have no effect on each other. What is the explanation?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

minotaurafe
Answered 2022-07-17 Author has 22 answers
Step 1
Let's compute E [ X Y ] E [ X ] E [ Y ] . We have E [ Y ] = 1 2 ( E [ Y T 0.5 ] + E [ Y T > 0.5 ] ) = 0.5 + 0.75 2 . Similarly, we have E [ X ] = 1 2 ( E [ X T 0.5 ] + E [ X T > 0.5 ] ) = 0.25 + 0.5 2 .
Then, we have E [ X Y ] = 1 2 ( E [ X Y T 0.5 ] + E [ X Y T > 0.5 ] ) . If T 0.5 , Y is independent from X, so E [ X Y T 0.5 ] = 0.25 0.5 . Similarly, when T > 0.5 , Y is indepdent from X, so E [ X Y T > 0.5 ] = 0.5 0.75 . Thus, we have cov ( X , Y ) = 1 2 ( 0.125 + 0.375 ) 1.25 0.75 4 0.0156 , indicating a positive correlation.
Not exactly what you’re looking for?
Ask My Question
Darian Hubbard
Answered 2022-07-18 Author has 7 answers
Step 1
Of course they are correlated. For one thing they are not independent (as X can only be bigger than 0.5 if Y > 0.5 and conversely Y can only be smaller than 0.5 if X 0.5 ).
But in fact they are positively correlated, as the cases with X > 0.5 , Y < 0.5 cannot happen.
Let’s calculate this:
E ( X ) = 0 0.5 x d x + 0.5 0 1 x d x = ( 0.5 2 + 0.5 ) / 2 = 0.75 / 2 = 0.375
and
E ( Y ) = 0.5 0 1 x d x + 0.5 1 x d x = ( 0.5 + ( 1 0.5 2 ) ) / 2 = 0.625
Then
E ( ( X 0.375 ) ( Y 0.625 ) ) = 0 0.5 ( x 0.375 ) 0 1 ( y 0.625 ) d y d x + 0 1 ( x 0.375 ) 0.5 1 ( y 0.625 ) d y d x
which amounts to 1 / 64 for the covariance. So the correlation is positive. We’d still need to calculate the variance of X, Y (which is the same) for the correlation:
E ( X 2 ) = 0 0.5 x 2 d x + 0.5 0 1 x 2 d x = ( 0.5 3 + 0.5 ) / 3 = ( 5 / 8 ) / 3 = 5 / 24
so
V a r X = 5 / 24 ( 3 / 8 ) 2 = 13 / 192
so we get a correlation of
( 1 / 64 ) / ( 13 / 192 ) = 3 / 13 0.23
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-03
Let the joint distribution of (X, Y) be bivariate normal with mean vector ( 0 0 ) and variance-covariance matrix
( 1 𝝆 𝝆 1 ) , where 𝟏 < 𝝆 < 𝟏 . Let 𝚽 𝝆 ( 𝟎 , 𝟎 ) = 𝑷 ( 𝑿 𝟎 , 𝒀 𝟎 ) . Then what will be Kendall’s τ coefficient between X and Y equal to?
asked 2022-05-28
A similarity/metric learning method that takes in the form of x T W y = z, where x and y are real valued vectors. For example, two images.
Breaking it into a more familiar form:
x T W y = i j w i j x i y j = z
This essentially looks very similar to polynomial regression with only interactions between features (without the polynomials). i.e.
z = f w ( x ) = i w i x i + i j = i + 1 w i j x i x j
I was curious to see if the optimization for the matrix W is the same as doing optimization for multivariate linear/polynomial regression, since x and y are fixed, and the only variate is the weight matrix W?
asked 2022-07-16
PCA vs Correlation
What is the relationship between (first) principal component(s) and the correlation matrix or the average correlation of the data. For example, in an empirical application I observe that the average correlation is almost the same as the ratio of the variance of the first principal component (first eigenvalue) to the total variance (sum of all eigenvalues).
Is there a mathematical relationship?
asked 2022-06-24
Let a sample ( x , y ) R 2 n be given, where y only attains the values 0 and 1. We can try to model this data set by either linear regression
y i = α 0 + β 0 x i
with the coefficients determined by the method of least squares or by logistic regression
π i = exp ( α 1 + β 1 x i ) 1 + exp ( α 1 + β 1 x i ) ,
where π i denotes the probability that y i = 1 under the given value x i and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.
Claim: If β 0 > 0 ( β 0 < 0), then β 1 > 0 ( β 1 > 0).
I figure this could be due to the sign of the correlation coefficient.
asked 2022-07-19
A consumer organization estimates that over a 1-year period 20% of cars will need to be repaired once, 6% will need repairs twice, and 2% will require three or more repairs. What is the probability that a car chosen at random will need repairs?
asked 2022-07-01
What I know, linear means polynomial of degree 1. But then, I found that in one of my lectures, the lecturers are saying that this regression is a linear regression:
Y i = α 0 + α 1 x i + α 2 x i 2
How is this a linear regression when it has quadratic terms in it?
asked 2022-07-18
Correlation: Concept to FormulaIn digital signal processing, we calculate the correlation between two discrete signals by multiplying corresponding samples of the two signals and then adding the products. Where does this process/formula for correlation come from?
I understand the concept of correlation (similarity) between two signals. But I fail to understand how it translates to the formula that it does.
All the texts I have seen so far start with this formula and explain cross correlation, auto correlation, etc. None of them attempt to explain how the formula was derived in the first place.

New questions

The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual racetracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The pro?t contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:
Max 200DRB + 280DRW
s.t.
20DRB + 25DRW 40,000 Steel Available
40DRB + 100DRW ? 120,000 Manufacturing minutes
60DRB + 40DRW ? 96,000 Assembly minutes
DRB, DRW ? 0
Optimal Objective Value = 424000.00000
Variable Value blackuced Cost
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 1000.00000 0.00000
DRW 800.00000 0.00000
Constraint Slack/ Surplus Dual Value
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 0.00000 8.80000
2 0.00000 0.60000
3 4000.00000 0.00000
Objective Allowable Allowable
Variable Coef?cient Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 200.00000 24.00000 88.00000
DRW 280.00000 220.00000 30.00000
RHS Allowable Allowable
Constraint Value Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 40000.00000 909.09091 10000.00000
2 120000.00000 40000.00000 5714.28571
3 96000.00000 Infnite 4000.00000
a. What are the optimal solution and the total profit contribution?
b. Another supplier offeblack to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
d. Because of increased competition, Deegan is considering blackucing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.
e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.