Question
Explain what a branching Markov chain is, and state clearly the Fundamental
Theorem for branching chains.
Amoeba reproduce by dividing into two. Some amoeba will die without
dividing. Assume that amoeba reproduce identically and independently and
Question
Describe what is meant by a Compound Poisson process.
Show that if A(z) is the probability generating function for the number of
events occurring at each point of the process, then the random variable X(t)
- the total number of events occurring i
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QUESTION The random variable R can take all non-negative integer
values
and its pgf is G(z). Show that the probability that R is an odd
integer is given be $\fraccfw_(1-G(-1)cfw_2$.\
An unbiased s
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QUESTION Transistors produced by a machine may be perfect,
slightly
damaged or unusable. 70\% of the production are perfect and 20\%
are slightly damaged. Let X be a variable giving the number of
QUESTION Write down th mgf of a Poisson distribution with mean .
Hence write down the mgf for the standardized Poisson variable. Show that
1 2
as this latter mgf tends to E 2 t , and hence confirm that a Poisson
variable can be approximated by a normal di
QUESTION The random variable R can take all non-negative integer
values and its pgf is G(z). Show that the probability that R is an odd integer
is given be (1G(1)
.
2
An unbiased sis-sided die is thrown repeatedly. Show that the pgf for the
z
.
number of
QUESTION Transistors produced by a machine may be perfect, slightly
damaged or unusable. 70% of the production are perfect and 20% are slightly
damaged. Let X be a variable giving the number of perfect transistors, Y
the number of slightly damaged transis
2
QUESTION The mgf of a random variable Y is e3t+8t . Prove that
E(Y)=3 and find Var(Y).
2 )2
2
+ . . . therefore =
ANSWER M (t) = e3t+8t = 1 + (3t + 8t2 ) + (3t+8t
2!
the coefficient of t = 3
E(X 2 )
= the coefficient of t2 = 8 + 92 therefore E(X 2 ) = 2
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QUESTION
A random variable X has pdf $f(X)=\fraccfw_kcfw_2e^cfw_-x(1+x)^2,\ -1 \leq x \leq
\infty$. Find k.
Show that the mgf of X is $ \fraccfw_e^cfw_-tcfw_(1-t)^3,\ t<1$. Hence
find Var(X). By e
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QUESTION
Find the mgf of the square of a standard normal variable. Hence
find the mgf of a $\chi^2$-distribution with $\nu$ degrees of
freedom. Hence show that the $\chi^2$-distribution is a
parti
QUESTION Find the mgf for a random variable which is uniform on
the interval a < x < b. Use the mgf to find the mean and variance of the
distribution.
1
a<x<b
ANSWER f (x) = ba
M (t) =
=
=
=
=
=
ext
dx
a ba
1 ext b
[
]
t ba a
ebt eat
t(b a)
1
(bt2 ) (bt)3
QUESTION A random variable X has pdf f (X) = k2 ex (1 + x)2 , 1
et
x . Find k. Show that the mgf of X is (1t)
3 , t < 1. Hence find Var(X).
By extending the results in an obvious way find also E(X )3 .
ANSWER f (x) = k2 ex (1 + x)2 1 x
k Z xt x
M (t)
=
Question
A simple random random walk has the set cfw_0, 1, 2, cdots, a 1, 1 as
possible states. States 0 and a are reflecting barriers from which reflection
is certain, i.e., if the random walk is in state 0 or a at step n the it will
be in state 1 or sta
Question
Explain what a branching Markov chain is, and state clearly the fundamental
theorem for branching chains. Suppose we begin with a single individual. Let
Fn (s) denote the probability generating function for the number of individuals in generation
Question
If a car driver has had k accidents in the time interval (0, t], then in the time
interval (t, t + t) she has a probability of
(i) ( + k)t + o(t) of having one accident,
(ii) 1 ( + k)t + o(t) of having no accidents,
(iii) o(t) of having more than
Question
A simple random walk can occupy states (0, 1, 2, ). For states j 2
there is a probability p of making a step of +1, and q of making a step of
-1, where p + q = 1 and pq 6= 0. State 0 is an absorbing barrier. State 1 is
partially reflecting in the
Question
(a) A Markov chain has the infinite transition probability matrix given below. Classify the states, justifying your conclusions. Find the mean
recurrence time for any positive recurrent states. (Label the states
1, 2, 3, in order.)
0 1 0
1
2
0
3
Question
Describe what is meant by a compound Poisson process.
Show that if A(z) is the probability generating function for hte number of
events occurring at each point of the process, then the random variable X()t
- the total number of events occurring i
Question
A gambler with initial capital z plays against an infinitely rich opponent. At
each play the gambler wins 2 with probability p, and loses 1 with probability
q = 1 p. Letting qz denote the probability that the gambler will eventually
be ruined, sh
Question
Explain what a branching Markov chain is. Suppose such a Markov chain
begins with single individual. Let A(s) denote the probability generating
function for the number of offspring of any individual. State how to use
A(s) to find the probability
Question
The telephone directory for Hirstville consists of three volumes, I, II, III.
The local Albanian take-away keeps is set of volumes in a neat pile by the
telephone. When a volume us consulted it is always replaced on the top of
the pile. The propr
Question
Explain what a branching chain is. Suppose a population is descended from
a single individual (generation 0). Let A(s) denote the probability generating
function for the number of offspring of any individual. Let Fn (s) denote the
probability gen
Question
Describe what is meant by a compound Poisson process. Show that if A(z)
is the probability generating function for the number of events occurring at
each point of the process, then the random variable X(t) - the total number
of events occurring i
Question
A gambler with initial capital z (where z is a positive integer) plays a coin
tossing game against an infinity rich opponent. Two fair coins are tossed: if
both show heads the gambler wins 2; if both show tails the gambler wins
1; otherwise the g
Question
(a) A Markov chain has three states, and transition probability matrix
0
1 0
P =
1p 0 p
0
1 0
where 0 < p < 1. Find the probability distribution for state occupancy
at the nth step (n 1) if initially all the states are equally likely to be
occu
Question
A population consists initially of b individuals. At any subsequent time
an individual in the population has, during any time interval of length t,
independent of previous history and of other individuals,
(i) a probability t + o(t) of producing
QUESTION Discrete random variables X and Y have the joint probability
function given in the following table.
Y\X 1
2
3
1
0.1 0.2 0
2
0.1 0.1 0.1
3
0.2 0 0.2
(i) Calculate E(X), E(Y), Var(X), Var(Y), Cov(X,Y).
(ii) Tabulate the probability function of Z=2Y
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QUESTION Discrete random variables X and Y have the joint
probability
function given in the following table.
\begincfw_tabularcfw_cccc
Y$\backslash$X&1&2&3\
1&0.1&0.2&0\
2&0.1&0.1&0.1\
3&0.2&0&0.2
QUESTION Eight golfers play a round of golf on two consecutive Saturdays.
On the first Saturday they returned scores of 72,89,69,70,85,71,96,86 and on
the second Saturday in the same order 72,80,71,70,82,72,90,84.
(a) Assuming that the differences in thei
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QUESTION The body-work of four-year old cars was compared for two
different models. Of 20 cars of the first model, 9 showed a
considerable degree of rusting, w
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QUESTION The life-time of a type of electric bulb can be assumed
to have an exponential distribution with mean
$\displaystyle\fraccfw_1cfw_\lambda$. 8 such bulbs are tested.
In each of the cases b
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QUESTION A random variable $X$ has pdf $f(x)=(\beta+1)x^cfw_\beta\ \
0 \leq x \leq 1,\ \ \beta<cfw_-1$. A random sample of $n$ values of
$X$, $x_1,x_2,.,x_n$ has been obtained.
\begincfw_descripti
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QUESTION Eight golfers play a round of golf on two consecutive
Saturdays. On the first Saturday they returned scores of
72,89,69,70,85,71,96,86 and on the seco
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QUESTION The diameters of ball bearings produced by a process are
normally distributed with standard deviation fo 0.04mm. A random
sample of 7 is taken and the