Transcribed TextTranscribed Text

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--

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    $75.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Advanced Statistics Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats