IE575 EXAM-II FALL 2013
MAX SCORE = 100 POINTS
One side of an 8.5”x11” formula sheet permitted; Calculator permitted
1. Discrete RV X is known to have the following transform:
2
8
6
1 1
1 1
1
1
1
𝑀𝑋 (𝑠) = 𝐾 ( + 𝑒 11𝑠 ) ( + e2𝑠 + 𝑒 4𝑠 ) ( e𝑠 + 𝑒 5𝑠 )
8 8
4 2
4
2
2
(a) Determine the numerical values of
(i)
K
(ii)
E[X]
(iii)
(iv)
(v)
δ𝑋2
𝑃𝑋 (0)
𝑃𝑋 (1)
For the remaining parts of this problem, you may use K, E[X], δ𝑋2 , 𝑃𝑋 (0), 𝑃𝑋 (1) as known
quantities. Do not substitute their numerical values.
(b) Discrete RV Y has the transform:
1
1 1 1
1
𝑀𝑌 (𝑠) = 𝑀𝑋 (𝑠) + ( + e𝑠 + 𝑒 2𝑠 )
2
2 3 3
3
Evaluate E[Y], δ2𝑌 , 𝑃𝑌 (1).
(c) Discrete RV V has the transform:
1
1
1
1
1
𝑀𝑉 (𝑠) = 𝑀𝑋 (𝑠) with e𝑠 replaced by (5 + 5 e𝑠 + 5 𝑒 2𝑠 + 5 𝑒 3𝑠 + 5 𝑒 4𝑠 )
Evaluate E[V], δ2𝑉 .
2. RVs X and Y have the following joint PDF:
1
𝑓𝑋,𝑌 (𝑥, 𝑦) = {2 𝑦
0
𝑓𝑜𝑟 0 ≤ x < 1 and 0 ≤ y < 2
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Event C is the event X>Y. It has probability P(C)=1/12.
(a) Determine and sketch the conditional PDF for X given C.
(b) Find the conditional expected value for X given both that C occurred and the
experimental value of Y is 0.5.
(c) Let T=max{X,Y} and find the PDF for T.
(d) Are X and Y independent? You need to substantiate your answer with appropriate
calculations.