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1. {a) The probability density function of a random variable X is g1ven by fx(.r) = · { CJ. O< .l'. < 2; O, otherwíse, where r ,s a constant. i. Find the value of c. ii. Find E(IX - IIJ. iii. Let Y be an independent and identically distributed copy of X. Find P(X -r Y > 2). (b) The probability mass function of X is given by μre I' Px(.r) = - 1 - , r = O, 1, . .•. . r. where μ > O is a constant. i. Show that the moment generating f unction M x ( t) of X is given by .\fx(t) = cxp(w' - 11), t E R. ÍI. Show that Var(XJ = μ. iii. Find E[.X <'xp(X + 1 ") , where Y is an independent and ldentically distributed copy of X. (Hint: Mx(t) = E(X l'.xp(tX)) .) (e) In a factory , there are three machines producing identical products . Toe time it takes for each machine to produce a product is exponent ially distributed , such that X. ~ Exp( ..\), 1 : 1, 2. 3, and the machines are mdependent of each other. Toe probability density function is given by i. Let T be the time unbl a product is produced by any of the 3 machines . Show that P(T > t) = ~-p (- 3-\t). What is the distribution of T? (Hint: T > t if and only if X, > t for ali 1 = 1, 2 3). il. lt was known that X:i > r. Show thal { f'X}>( -2,\l) . P(T > t lX2 > e) = rxp (- 3,\t + ,\e). O< l < e:; iii. Find E(TIX2 > r) . You can use, for any positive random variable Y, E(l") = ¡"GP (Y > y )dy . ------- 2. The joint probability density f unctíon of \' and Y is given by Let (a) Show that (b) Show that (e) Show that a.r2y2 fx ,y(x, y) = j :i 2 • O< r < y < l. r +y 1 O < U < v'2 and o < V < l. 1 Q < 11 < r..• v2 1 o< u< -12· O< t' < l. and (d) Find the value of a. Hint: use integration by parts on the left hand side of ¡1/./2 1 --.,du. = log(l + V2). 0 1 - u4. In a particular emergency scenario, an ambulance arrives in time with probability 0.9. After arriving , the arnount of time T the paramedics spent before leaving the scene has probabihty density function (t > O} { 2,<~'. ambulance arrives in time; fr(l) = 1, .- i,. otherwise. (b) One observes that the paramedics spent less than 1 unit of time befare leavIng. What is the probabil ity they arrived in time? (e) Le1 X be the time used for the pararnedics to return to their hospital. The probability density of X is given by (.r > O) T< e; T 2: e, where c. > O. Let .1 be the event that the ambulance arrives in time. i. Rnd P(X > e n T < e n .4) and P(X > e n T? e n .4). ii. By conditioning on T < e or T ~ e only, show that iii. Using parts i and ii, show that !) - 9r.-k + !)r.- k P(Ambutance arrives in timelX >e)= 10 !) _.,. e _... 'e - e '"' + oc .... + e--

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