If the joint probability density of X and Y is given by f(x, y)=2 for x>0, y>0, x+y<1, and f(x, y) = 0 elsewhere, find (a)P(X<=1/2, Y<=1/2); (b)P(X+Y>2/3); (c)P(X>2Y)
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If the joint probability density of X and Y is given by f(x, y)=2 for x>0, y>0, x+y<1, and f(x, y) = 0 elsewhere, find
(a)P(X<=1/2, Y<=1/2);
(b)P(X+Y>2/3);
(c)P(X>2Y)
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- 1. A continuous random variable X is defined by (3+x) f(x) - 3 sxs- 1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean IIIf the joint probability density of X and Y is given by f(x, y) =⎧⎪⎨⎪⎩13(x + y) for 0 < x < 1, 0 < y < 20 elsewherefind the variance of W = 3X + 4Y − 5.1. A continuous random variable X is defined by (3+x) f(x) - 3sxs-1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean 2. If the probability density function of random variable is given by f(x) = wch 1srs2 sech x a. Find the mean b. Find the total area 3. If the probability density function of random variable is given by f(x) = sinh 2x 1sxs2 a. Find the mean b. Find the total area c. Illustrate the graph (use bell shaped figure) II II
- There are only two states of the world, when a person is well with probability (1-p) and ill with probability p, where (1-p)= 1/3 and p = 2/3. Consider Adam who has utility function U = (Y1, Y2,1 – p,p) = Y,"-P)Y?, where Y; is the income and i =1 is well and i = 2 is ill. When Adam is well, he earns $1000, but when he is ill, he losses $300 in health expenditures and earnings such that Y2 = $700. What is the maximum total premium that Adam is still willing-to-pay? (1-р).The life lengths of two transistors in an electronic circuit is a random vector (X; Y) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y) is given by x 2 0, y 2 0 fx.,fx.v) = 20 else Then the probability that the first transistor burned during half hour given that the second one lasts at least half hour equals Select one: a. 0.606 b. 0.3935 C. 0.6318 d. 0.3669 e. 0.7772Suppose that therandom variable X has the probability density function c(1– x2) for -1s x <1 f(x) elsewhere What is the variance of X
- Find k if the joint probability distribution of X, Y, andZ is given by f(x, y, z) = kxyzfor x = 1, 2; y = 1, 2, 3; z = 1, 2.1) The probability density function of the weight in kilogram of passenger baggage 50 received by an airline is f(x) = ; for 0 < x < 50. The free baggage allowance of an airline is 49x2 up to 32 kilograms. What is the probability that a checked baggage is free?9) The effectiveness of solar-energy heating units depends on the amount of radiation available from the sun. During a typical October, daily total solar radiation in Tampa, Florida, approximately follows the following probability density function (units are in hundreds of calories): 3 f (x) ={32 (x- 2)(6-x) for2Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20IF it was XG (23) Independent of Xen G (13) Find Miley D-Probability density Function of the Variable Y-X+Xq 2 -the Mean variance of the variable
