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Suppose that two random variables X1 and X2 have a continuous joint distribution for which the joint p.d.f. is as follows:
f(x1,x2) = x1 + x2 for 0 < x1 < 1 and 0 < x2 < 1,
(and 0 otherwise)
Find the p.d.f. of Y = X1 * X2
If anyone could help, it would be greatly appreciated!
Shown below are the number of galleys for a manuscript (X) and the dollar cost of correcting typographical errors (Y) in a random sample of recent orders handled by a firm specializing in technical manuscripts. Assume that the regression model Yi = (β1*Xi) + εi is appropriate, with normally distributed independent error terms whose variance is σ^2 = 16.
(Note that above, Xi means X sub i and the same goes for εi and Yi.)
The (Xi, Yi) are as follows:
1: (7, 128)
2: (12, 213)
3: (4, 75)
4: (14, 250)
5: (25, 446)
6: (30, 540)
What is the likelihood for the six Y observations for σ^2 = 16?
I know that in order to obtain the likelihood, I have to find the density of each observation, which is given by the equation:
1/(σsqrt( 2π))* exp(-1/2 ((Yi-β0- β1Xi)/ σ)
and then multiply them.
But what values do I have to use for β0 and β1?
Thanks for any help!
Let's say you are fitting a linear regression function to a set of data.
Let's say you get a positive relation when plotting the residuals against the actual outcomes and no relation when you plot the residuals against the fitted values. In what event would this difference occur? Also, which would be the more meaningful plot?
Thanks, mathsyperson! Apparently there was a problem with my textbook and it was supposed to include q, which is where I got confused.
A scientist is about to compile a series of 11 similar programs. Suppose A_i is the event that the ith program compiles successfully for i = 1,...,11. When the programming task is simple, the scientist expects that 80% of programs will compile. When the programming task is hard, he expects that 40% of the programs will compile. Let B be the event that the programming task is simple and B complement be the event that the programming task is hard. The scientist believes that the events A_1,...A_11 are conditionally independent given B and given B complement. Let A be the event that exactly eight out of eleven programs compiled. What is the conditional probability of B given A?
Here's what I gather so far:
Conditional on B - The conditional probability that a particular collection of 8 programs out of 11 will compile is
(0.8)^8 * (0.2)^3 = 0.001342.
There are 11 choose 8 such collections of eight programs out of the 11. This is equal to 165, so the probability that exactly 8 programs will compile is 165 * 0.001342 = 0.2215.
Conditional on B complement - The conditional probability that a particular collection of 8 programs out of 11 will compile is
(0.4)^8 * (0.6)^3 = 0.0001416. This times 165 gives the probability of exactly 8 programs compiling. This value is equal to 0.02335.
I'm confused as to what to do to find the conditional probability of B given A, though. If anyone could help, it would be greatly appreciated!
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