Engineering-All six faces of a fair die have equal probabilities

Engineering-All six faces of a fair die have equal probabilities

Subject: Engineering    / General Engineering
Question
1. All six faces of a fair die have equal probabilities of appearing. Suppose you suspect the face with the six dots does not appear with equal probability. You toss the die 180 times trying to detect any irregularity.
a. If the die is fair, what is the ensemble average of the number of times the six dots will appear?
(1 pt)
b. Of course, in practice the actual number of times the six dots will appear will not be exactly equal to that ensemble average. Write down the analytical express for the probability that a fair die will have less than 20 and and more than 40 number of six dots appearing in 180 tosess? Do not evaluate this expression.
(2 pts)
c. Since the above analytical expression of the probability is too computationally complicated to evaluate, let us use the Central Limit Theorem to approximate this expression. What is the numerical value for this aproximate expression? Hint: We know for a Gaussian CDF, (0) = 0.5; (1) = 0.8413; (2) = 0.9772 . (3 pts)
2. Let {X1 , X2 , X3 } be a zero-mean random sequence with an autocorrelation function
R(Xi Xj ) = E{Xi Xj } = a|ij| , i, j = 1, 2, 3, where 0 < a < 1.
a. Under the mean-square (ms) error criterion, we vary {c1 , c2 } such that 2 (c1 , c2 ) =
E{(X3 (c1 X1 + c2 X2 ))2 } is minimized with 2 (1 , c2 ) = min{2 (c1 , c2 )} =
min c
E{(X3 (1 X1 + c2 X2 ))2 }. Find {1 , c2 }. (2 pts). Find 2 (1 , c2 ). (2 pts)
c
c
min c
b. Under the mean-square (ms) error criterion, we vary {c1 } such that 2 (c1 ) = E{(X2 c1 X1 )2 } is minimized with 2 (1 ) = min{2 (c1 )} = E{(X2 (1 X1 )2 }.
c
min c
Find {1 }. (2 pts). Find 2 (1 ). (2 pts)
c
c
min
3. Let X and Y be two independent positive-valued r.v.s with their pdfs given by fX (x) = 0, x < 0 and fY (y) = 0, y < 0. Dene the r.v. Z = X Y, where is the regular multiplication (e.g., 3 2 = 6.) and is not the convolution operation. Find
the pdf of fZ (z), 0 < z < . Hint: Start with FZ (z).
(6 pts)
4. Let the random vector X = [X1 , X2 ]T have a mean vector X = [1, 2]T and a covariance
matrix
[
]
1 1
RX =
.
1 2
Let Y = AX, where

[

A=

1 2
1 3

]

.

Find the mean vector Y and the covariance matrix RY of Y.