1. Suppose that the universal set S is defined as S = cfw_1, 2, , 10 and A = cfw_1, 2, 3,
B = cfw_x S : 2 x 7, and C = cfw_7, 8, 9, 10.
(a) Find A B
(b) Find (A C) B
(c) Find A (B C)
(d) Do A, B, and C form a partition of S?
A B = cfw_1, 2,
1. Let X be the number of the cars being repaired at a repair shop. We have the
- At any time, there are at most 3 cars being repaired.
- The probability of having 2 cars at the shop is the same as the probability of
having one car.
1. A coffee shop has 4 different types of coffee. You can order your coffee in a small,
medium, or large cup. You can also choose whether you want to add cream, sugar,
or milk (any combination is possible, for example, you can choose to add all three).
1. (25%) Suppose that we have three events A, B, and . We know
(5%) Are A and C disjoint? Are they indep ndent
(5'Yo) Find peA n B n C).
1. (15%) Suppo e thatI give a five~point multiple~choice exam and
the resulting core X has the following probability mass function
e figur d outthat
0.1 k;:; 0
0.2 k = 1.
P (k)= 0.2 , k = 2:
lot the cumulative distribution fun
J. (15%), Let X be a di$crete random variable .
With the following .probabil ty mass f LmCbon
k ;: 5
(a) (5%) 'lot the cumulative d' . .
d E[X] th
lstributlon function (CD
1. Let X be a discrete random variable with the following PMF
for x = 0
for x = 1
PX (x) =
for x = 2
(a) Find RX , the range of the random variable X.
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|
Chapter 4, Problem 13: Let X be a random variable with the following CDF:
for x < 0
for 0 x < 14
FX (x) =
x + 12
for 14 x < 21
for x 21
1. Plot FX (x) and explain why X is a mixed random variable.
2. Find P (X 13 ).
3. Find P (X 14 ).
4. Write CDF
1. Let X be a continuous random variable with the following PDF
fX (x) =
where c is a positive constant.
(a) Find c.
(b) Find the CDF of X, FX (x).
(c) Find P (2 < X < 5).
(d) Find EX.
To find c
1. Consider two random variables X and Y with joint PMF given in Table ?.
Table 1: Joint PMF of X and Y in Problem prob:table-cov
Find Cov(X, Y ) and (X, Y ).
First, we find PMFs of X and Y :
RX = c
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The IQ of randomly selected individuals is often assumed to follow a
l dist ib ti
100 d st d d d i ti 15
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If we are to perform r experiments in order such that there are n1 possible
outcomes of the first experiment, n2 possible outcomes of the second experiment,
, nr possible outcomes of the rth expe
Unordered sampling without replacement:
There are n distinguishable objects; we want to choose k
objects but ordering does not matter:1,2,3 =3,2,1=2,3,1.
Thus the number of k-combinations of n objects is:
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The number of ways