Problem 1 Suppose that X and Y are jo intly continuous randon1 variables, X rv Exponential(!) and the conditional density of Y given X = x is f r 1 x (y Ix) = e–(y–x) , y > x (in other words, an Exponential( ) random variable shif ted x units to the right).
1 pt (a) Calculate the joint probability density function of X and Y , being sure to clearly specify the valid domain of points (x , y ).
2 pt s (b) Calculate the marginal probability density function of Y . Nan1e the marginal distribution of Y as a known distribution and specify any parameter values.
1pt s (c) What is the conditional distribution of X given that Y = y ?
2 pt s (d) Compute the expected value of X Y .
3pt s (e) Using the above inforn1ation , give the values of JEX , lEY , Var(X), Var(Y ) and Cov(X, Y).
Proble1n 3 Goldilocks breaks into a house and samples the three bowls of porridge that are sitting on the table. Each bowl is at a randon1temperature uniformly distributed between 0°C and l00°C, independently of the others. Goldilocks considers that the hottest bowl will be too hot for her, the coolest bowl too cold for her, and the third (middle) bowl will be ju st right. Let H, C, J be the temperatures (in °C) of the hottest, coldest and “j ust right” bowls respectively.
4 pt s (a) Calculate the probability density functions of H , C and J . They are all scaled version s of wellknown distributionsname the corresponding distributions (specifying any parame ter values).
1 pt (b) Give thejoi nt density function of ( H , C), specifying the domain carefully.
1 pt (c) Calculate the expectation of the range of ten1peratures an1ong the porridge bowls (i.e.,
H – C).
3pt s (d) Calculate the standard deviation of H – C .
Proble1n 2 Let Z 1, Z2 , . . . , Zk , Zk+ i be independent and identically distributed standard norn1al random variables.
1 pt  (a)  What is the distribution of Z? + Zi + ···+ Zf ? 
2 pts  (b)  Hence or otherwise compute the probability density function of the random variable 
R := J Zi + Zi + ···+ Zf .
1 pt (c) What is the distribution of Z+ i ? Express it as a scaled version of a known distribution.
Proble1n 4 A randon1variable T taking only positive integer values is said to be memoryless if T has the same distribution as the rangerestricted distribution of T – 1conditional on the event T > 1. If you think of T as the random waiting time until a certain event happens, T is memoryless if, given that the event has not happened at time 1 (the first time step), the conditional distribution of the
remaining waiting time T – 1is the same as the unconditional distribution of T .
2 pts  (a)  Let p = IP'{T = 1}. Prove that for any positive integer s,  
IP'{T > s} = (1 – p) IP'{ T > s – l}.  
1pt1 pt2 pts  (b)(c)(d)  Hence write down an expression in tern1s of p for IP'{T > s }.Hence or otherwise show that T r._, Geometric(p).Write down the c.d.f. Fr (x) of T , i.e., give a formula for Fr (x ) valid for every x  > 0. In 
order to deal with all x values, you should make use of the floor function Lx J which means
the greatest integer n such that n < x .
T,
2 pts (e) For each positive integer n, define random variables Tn r._, Geometric(p / n) and Un = !!_ .
n

Write down the c.d.f. Fu
(x) of Un.
2 pts (f) Write down a formula for F (x ) = limn+oo Fun (x). What distribution is F the c.d.f. of?
You may use without proof the wellknown limit
lin1 1 +c ) LnxJ
= ecx
for any real constants c and x .
n+oo ( n
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