Question

Final2.jpeg iat probability density function is given by f(x, y (2(x2+ y'), 0<x <2, 0cy3 Find the marginal densities of X and Y. Lip 2 Are X and Y independent? 4p c. Find probability X + Y < i , P(X+Y < 1 ). The amount of time, in hours, that a computer functions before breaking down .b] uniformly distributed on [o is continuous random variable T

question 4, 3 sub-questions

Answer 1

a)

$f(x) = \int _0^3\frac{1}{26}\left(x^2&plus;y^2\right)dy=\frac{1}{26}\left(3x^2&plus;9\right)$

$f(y) =\int _0^2\frac{1}{26}\left(x^2&plus;y^2\right)dx=\frac{1}{39}\left(3y^2&plus;4\right)$

b)

X and Y are not independent as f(x,y) is not equal to f(x)*f(y)

c)

P(X + Y < 1) =

$\int _0^1\int _0^{1-y}\frac{1}{26}\left(x^2&plus;y^2\right)dxdy=\frac{1}{156}\quad \left(\mathrm{Decimal:\quad }\:0.00641\dots \right)$

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