f(x, y) = ce−(2x2+6xy+3y2), −∞ < x < ∞, −∞ < y < ∞
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Can the following be a density
f(x, y) = ce−(2x2+6xy+3y2), −∞ < x < ∞, −∞ < y < ∞
where c is a positive constant.
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- X and Y are jointly continuous with joint pdf f (r, y) = { cxy if 0 X) (c) Find marginal pdf's of X and of Y (d) Are X and Y independent? (?? Given: We start (as always!) by drawing the support set. (See below, left.) ソテX 1 Blue: subset support set of support set with y>x 1 0.5 1 x y=1-x Hints for (a): Range of y is from 0 to 1 – x Range of x is from 0 to 1 Hints for (b): Range of y is from x to 1 – x Range of x is from 0 to 0.5 -A programmer plans to develop a new software system. In planning for the operating system that he will use, he needs to estimate the percentage of computers that use a new operating system. How many computers must be surveyed in order to be 90% confident that his estimate is in error by no more than three percentage points? Complete parts (a) through (c) below. a) Assume that nothing is known about the percentage of computers with new operating systems. n = (Round up to the nearest integer.) b) Assume that a recent survey suggests that about 98% of computers use a new operating system. n= (Round up to the nearest integer.) c) Does the additional survey information from part (b) have much of an effect on the sample size that is required? O A. Yes, using the additional survey information from part (b) dramatically increases the sample size. O B. No, using the additional survey information from part (b) only slightly increases the sample size. OC. No, using the additional survey…Consider the function fy (x,y)=c(x² +y² ), a joint probability density function over 02.5 and Y <1.5) Are X and Y independent? You must demonstrate this using the definition. What is the correlation between X and Y? е. f. g. What is the probability that Y<1.5 given X=0?
- In the picture below, what can you infer? Note: we are going to order according to the light blue area, which is slightly to the left of the average demand. Area = Pe=C₂! (C₂+₂) = prob of being "over" z (# σ from μ) -1 of σ from H. · z=norm.s.inv(P.) μ= expected demand 1 = std dev in demand 2 z (#from μ) itu Co Q=zo+μ - On this scale put the actual numbers specific to the problem You are more likely to have some product left over than run out. Cost of understocking must have been higher than cost of overstocking. Demand will be always be below average.. You are more likely to run out of product than have some left over.3. Find E(1+) for the density function 3 f(x, y) = 11 (x3 -3x²+2x)(y² - 3y+3), 0, 1 < x < 3,0 < y < 1, elsewhere) Find an analytic function f(z) = u(x, y) | iv(x, y) whcrc u(x, y) = c²x cos(y) c²ysin(y). %3|
- Suppose X has PDF fx(x) = x/2 for 0 < x < 2 and 0 otherwise. (a) Find E[X] for n = = 0, 1, 2,...,. (b) Find the density function of Y differentiating. = X(2X) by computing P(Y ≥ y) and then5. The joint density function is given by [30.xy², x-1 0| X = 0.75).1.5 Solve the below problem: Let Y, and Y, have a joint density function given by f(n. Y2) = $3y1 0< y2 < yı <1 lo, elsewhere. The marginal density function for Y, is f(v2) =(1 – y3), 0 < y2 < 1. Find the marginal density function of Y1. (NB: Show all your calculations)
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