If E [ X ] = 75 E [ Y ] = 75 V a r ( X ) = 10 var ( Y ) = 12 cov ( X , Y ) = − 3 give an upper bound to a. P { | X − Y | > 15 } ; b. P { X > Y + 15 } ; c. P { Y > X + 15 } .
If E [ X ] = 75 E [ Y ] = 75 V a r ( X ) = 10 var ( Y ) = 12 cov ( X , Y ) = − 3 give an upper bound to a. P { | X − Y | > 15 } ; b. P { X > Y + 15 } ; c. P { Y > X + 15 } .
Let (Ω, F, P) be a probability space and let X : Ω → R be a randomvariable whose probability density function is given by f(x) = 12 |x|e−|x| forx ∈ R.(i) Find the characteristic function of the random variable X.[8 Marks](ii) Using the result of (i), calculate the first two moments of therandom variable X, i.e., E(Xn) for n = 1, 2. [6 Marks]Total marks 16 (iii) What is the variance of X?
ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rule
Let X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .
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