probability, random processes and inference
TRANSCRIPT
INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION
Probability, Random Processes and Inference
Dr. Ponciano Jorge Escamilla Ambrosio [email protected]
http://www.cic.ipn.mx/~pescamilla/
Laboratorio de
Ciberseguridad
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1. Choose two numbers X1, X2 without
replacement and with equal probabilities from
the set {1, 2, 3}, and let X = max{X1, X2} and
Y = min{X1, X2}. a) Find the joint PMF of X
and Y; b) find the marginals of X = max(X1,
X2) and Y = min(X1, X2) from the joint PMF.
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Problem 1
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Solution Problem 1
X1 1 1 2 2 3 3
X2 2 3 1 3 1 2
X 2 3 2 3 3 3
Y 1 1 1 2 1 2
Y\X 2 3 X Marginal
1 1/3 1/3 2/3
2 0 1/3 1/3
Y Marginal 1/3 1/3 1
The possible combinations are:
a) PMF y b) son:
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2. You just rented a large house and the realtor
gave you 6 keys, one for each of the 6 doors of
the house. Unfortunately, all keys look identical,
so to open the front door, you try them at
random. Find the PMF of the number of trials
you will need to open the door under the
assumption that after an unsuccessful trial, you
mark the corresponding key, so you never try it
again.
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Problem 2
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Solution Problem 2
𝑃 𝑋 = 1 =1
6
𝑃 𝑋 = 2 =5
6∙1
5=1
6
𝑃 𝑋 = 3 =5
6∙4
5∙1
4=1
6
𝑃 𝑋 = 4 =5
6∙4
5∙3
4∙1
3=1
6
𝑃 𝑋 = 5 =5
6∙4
5∙3
4∙2
3∙1
2=1
6
𝑃 𝑋 = 6 =5
6∙4
5∙3
4∙2
3∙1
2∙1
1=1
6
𝑃𝑋 𝑥 =
1
6para 𝑋 = 1, 2, 3, 4, 5 , 6
0 cualquier otro caso
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3. Accidentally, two depleted batteries got into a
set of five batteries. To remove the two depleted
batteries, the batteries are tested one by one in a
random order. Let the random variable X denote
the number of batteries that must be tested to
find the two depleted batteries. What is the
probability mass function of X?
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Problem 3
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Solution Problem 3
To find the two depleted batteries, you need at least two tests but no more than
four tests. Label the batteries as 1, 2, . . . , 5. Think of the order in which the
batteries are placed for testing as a random permutation of the numbers 1, 2, . . . ,
5. The sample space has 5! equally likely outcomes. You need two tests if the first
two batteries tested are depleted. The number of outcomes for which the two
depleted batteries are on the positions 1 and 2 is 2 × 1 × 3!. Hence:
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Solution Problem 3
You need three tests if the first three batteries tested are not depleted or if a
second depleted battery is found at the third test. This leads to:
The probability P (X = 4) follows from P (X = 4) = 1 − P (X = 2) − P (X = 3) =
6/10.
Alternatively, the probability mass function of X can be obtained by using
conditional probabilities. This gives
P (X = 0) = 2/5 × 1/4 = 1/10 and P (X = 2) = 3/5 × 2/4 × 1/3 + 2/5 × 3/4 × 1/3 +
3/5 × 2/4 × 1/3 = 3/10.
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4. You roll a fair dice twice. Let the random variable X be the
product of the outcomes of the two rolls. What is the probability
mass function of X? What are the expected value and the
standard deviation of X?
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Problem 4
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Solution Problem 4
The random variable X is defined on a sample space consisting of the 36
equiprobable elements (i, j), where i, j = 1, 2, . . . , 6. The random variable X
takes on the value i × j for the realization (i, j). The random variable X takes
on the 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, and 36 with
respective probabilities 1/36, 2/36, 2/36, 4/36, 2/36, 4/36, 2/36, 1/36, 2/36,
4/36, 2/36, 1/36, 2/36,‘ 2/36, 2/36, 1/36, and 1/36 . The expected value of X is
easiest computed as:
Also,
Hence E(X) = 12.25 and σ(X) =
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5. Bobo, the amoeba, currently lives alone in a
pond. After one minute Bobo will either die,
split into two amoebas, or stay the same, with
equal probability. Find the expectation and
variance for the number of amoebas in the pond
after one minute.
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Problem 5
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Problem 5
Let X denote the number of amoebas after one minute, then by definition of
expected value:
E[X] = 0 · P (X = 0) + 1 · P (X = 1) + 2 · P (X = 2)
= 0 + 1/3 + 2/3 = 1
To compute the variance first evaluate the second moment E[X2] (or use the
definition directly):
E[X2 ] = E[X] = 0 · P (X = 0) + 1 · P (X = 1) + 4 · P (X = 2)
= 0 + 1/3 + 4/3 = 5/3
Thus,
Var[X] = E[X2 ] - (E[X])2 = 5/3 - 1 = 2/3
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6. In a digital communication system a "0" or "1" is
transmitted to a receiver. Typically, either bit is equally
likely to occur so that a prior probability of 1/2 is
assumed. At the receiver a decoding error can be made
due to channel noise, so that a 0 may be mistaken for a
1 and vice versa. Defining the probability of decoding a
1 when a 0 is transmitted as ϵ and a 0 when a 1 is
transmitted also as ϵ, what is the overall probability of
an error?
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Problem 6
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Solution Problem 6
The effect of the channel is to introduce an error so that even if we know
which bit was transmitted, we do not know the received bit.
The probability of error is:
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7. A radioactive source emits particles toward a Geiger
counter. The number of particles that are emitted in a
given time interval is Poisson distributed with expected
value λ. An emitted particle is recorded by the counter
with probability p, independently of the other particles.
Let the random variable X be the number of recorded
particles in the given time interval and Y be the number
of unrecorded particles in the time interval. What are
the probability mass functions of X and Y ? Are X and
Y independent?
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Problem 7
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Solution Problem 7
Let the random variable N be the number of particles emitted in the given
time interval. Noting that P (X = j, Y = k) = P (X = j, Y = k, N = j + k) and
using the formula P (AB) = P (A | B) P (B), it follows that:
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Solution Problem 7
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8. Let X be a random variable with PMF:
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Problem 8
(a) Find a and E[X].
(b) What is the PMF of the random variable Z = (X - E[X])2 ?
(c) Using the result from part (b), find the variance of X.
Find the variance of X using the formula var(X) = 𝑥 − 𝐸 𝑋 2𝑥 𝑝𝑋 𝑥 .
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Solution Problem 8
(a) The scalar a must satisfy:
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Solution Problem 8
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9. Alvin throws darts at a circular target of
radius r and is equally likely to hit any point in
the target. Let X be the distance of Alvin’s hit
from the center.
(a) Find the PDF, the mean and the variance of
X.
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Problem 9
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Solution Problem 9
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10. A person playing darts finds that the
probability at which the dart falls between r and r
+ dr is:
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Problem 10
where, R is the distance of the impact from the center of the dartboard, c is a
constant and a is the radius of the dartboard. Find the probability of hitting a
target of radius b concentric with the dartboard (see figure below). Assume also
that impact on the dartboard is always done.
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Solution Problem 10