Induction Hypothesis: Assume that
Induction Step: We need to show that
13(5) >15) =ZBMOJ) for r: k+ 1.
00 Ak'H miceA1
P(Sk+1 > t) = / de.
Use an integration by parts with 'U. : AkH
1. [10 points] Consider a small store that receives 4 copies of a local
newspaper daily. Suppose that the customers that would like to purchase
the newspaper can be modeled as a Poisson process with a rate of 3
customers per day. We assume that copies of
. The time waiting in the emergency room of a hospital before one is
called to see a doctor often seems to be exponentially distributed with
mean equal to 60 minutes. Let X be the rand
2. [10 points] Let X have a geometric distribution with parameter 39. Its
probability mass function is
(a) Using the above probability mass function, Show that X has the
GX(z)=1_(f:W IZI <1/(1p).
(b) Use the
3. [10 points] When the production of the transistors is under statistical
quality control about 2% of the transistors are going to be defective.
Suppose that a transistor will be defective independently of the defec-
tive status of the other transistors.
1. [10 points]
(a) We want P(N(1) 2 4) = 1 23:0 cfw_es/51).
b Let X be the number of news a ers sold on a articular da . Its
P l3 p 3
probability mass function is
P(N(1) = 0) = e_3(3/01) J: 0
P(N(1) = 2) = e_3(31/11), :r 2
mm) : Pom) : 2) : 5382/21), x e
Exercise 5.3.4: Consider a discrete random variable X whose probabil-
ity generating function is given by
Observe that when 2 = 1, Gx(z) = 1. Find the probabilities px(0), px(1), px(2), px(3),
Suggested Exercises - Chapter 5
Exercise 5.3.3: Let X be a discrete random variable with probability
mass function given by
1/10, x = 1
2/10, 59 = 2
px(a:) = 3/10, 55' = 3
4/10, 55 = 4
Find the probability generat
1. Consider the following sample space 9 = cfw_(1, 6, ad, e. Is .7: a a-eld,
.7: = cfw_[3, cfw_(L, b, cfw_a, cfw_b, cfw_c, cfw_C,d,, cfw_51, b, 0,151, e?
2. Let Q be a sample s
9 9 T A A E GD) &
 0. 69424382 0. 03661395 0.32462651 0. 67352867 048179223
Use the inverse transformation technique to convert this sample into a
random sample from an exponential distribution with rate A = 0.1.
Let X have an exponential di
R(t) = P(X>t)
P(X>s) _ 3(5)
Therefore, R(s +15) = R(5) R(s).
. Let U = F(X). We know that U N Um, 1). Note that P takes values
between 0 and 1. Let us nd its c.d.f. (for 0
Recall: nbinom is the R name for the modied negative binomial.
Solution using formulae:
(a) We want
P(X > 2) = 1 - P(X s 2) = 1 i (5:) (0.02)i(0.98)50i,
where X m B(50,0.02).
(b) Let p : P(X > 2) be the probability that a sample will give an
4. [10 points] Consider a collection of 18 components. Assume that 5 of
these components are defective. We select 2 of the components without
replacement. Dene D:- as the event that the 2th selected component
For this question, please comput
 4.741221 8.212677 23.581586 15.577463 6.189225
R uses the inverse transformation technique to generate these values
from a random sample of size n from a U(0, 1) distribution. We give
you a sample of size n = 5
Universit dOttawa - University of Ottawa
Facult des sclences Faculty of Science
Mathmuthues e1. de statisthue Mathematics and Statistics
MAT 4371 SYS 5120
February 25, 2016 Professor G. Lamothe
Duration: 80 minutes
This 2'5 a
1. We have cfw_a E J' and cfw_c E .F. So we must have cfw_o,c = cfw_a U cfw_c E
.7, since a arr-eld is closed under unions. But, cfw_(1, C F. This means
that J: is not a U-eld.
2. We know that J: is a nonempty
5. [10 points] The joint probability mass function for (X, Y) is given in
the following table. For this question, please compute the nal numer-
ical answer for each part.
17 y PXYUCJU)
1 1 1/4
1.5 2 1/8
1.5 3 1/4
2.5 4 1/4
3 5 1/8
(a) Determine the follow
a e T A A e e a
Exercise 5.3.5: Let X be a discrete random variable with the following
probability mass function
Mk) =P(X =k) = ( 2 ) pk(lp)_k,k =0,1,.,n,
where 0 < p < 1. This distribution is referred to as the binomial distribution
with parameters 71 an