## Transcribed 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--

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