In: Math
John invited four friends to his birthday party and his friends will arrive independently of one another. The arrival time x for each friend is uniform over (0; 4) and these arrival times are i.i.d. Answer to the following questions:
(a) the density of the arrival time who arrives at second.
(b) the probability that the second person to arrive between 0 and
2.
(c) the expected arrival time of the second person to arrive.
Let be
the arrival time of 4 friends which are uniformly distributed over
(0,4).
The pdf is
The cdf is
Let
be the ordered statistics such that
. That is
is the arrival time of ith person to arrive
the pdf of is
a) The pdf for i=2
ans: the density of the arrival time who arrives at second is
b) the probability that the second person to arrive between 0 and 2 is
ans: the probability that the second person to arrive between 0 and 2 is 0.6875
c) The expected value of
is
ans: the expected arrival time of the second person to arrive is 1.6