Let X be a ternary rv taking on the three values 0, 1, 2 with probabilities p0, p1, p2 respectively. Find the median of X for each of the cases below.
(a) p0 = 0.2, p1 = 0.4, p2 = 0.4.
(b) p0 = 0.2, p1 = 0.2, p2 = 0.6.
(c) p0 = 0.2, p1 = 0.3, p2 = 0.5.
Note 1: The median is not unique in (c). Find the interval of values that are medians. Note 2: Some people force the median to be distinct by defining it as the midpoint of the interval satisfying the definition given here.
(d) Now suppose that X is non-negative and continuous with the density fX (x) = 1 for 0 ≤ x ≤ 0.5 and fX (x) = 0 for 0.5 x ≤ 1. We know that fX (x) is positive for all x > 1, but it is otherwise unknown. Find the median or interval of medians.
The median is sometimes (incorrectly) defined as that α for which Pr{X > α} = Pr{X α}. Show that it is possible for no such α to exist. Hint: Look at the examples above.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.