Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 PROBABILITY AND STATISTICS I, FALL 2010
EXAMPLE CLASS 4 Distribution
Elements of Theory
Abbreviated Event Notation
Usually we abbreviate the description of the event/set
to simply as
so that we can also
abbreviate the probability notation
as
or even simpler as
. This is primarily
because the two probability measures
and are consistent due to the property of the random variable .
Definition of CDF:
,
Properties of CDF:
1.
2.
3. , if 4.
5.
6.
7.
8. , Definition of PMF:
(Using the abbreviated event notation) where are the points at which of a discrete random variable Properties of PMF:
1.
2.
3. ∑
4. If ,
, if
∑ is a continuous random variable, then . jumps. Definition of PDF: Properties of PDF:
1.
2. ∫
3. ∫ 4. ∫ Gamma Function and Beta Function:
, ∫ .
. ∫ Exercises
1. The Random Experiment is “Tossing a coin (not necessarily a fair one) once. And if the coin turns out a head, then
you earn 1 dollar, otherwise, 0 dollar.” You are interested in how many dollars you will win after tossing it once.
Define an appropriate random variable to describe the experiment whose state space should be {0,1} and derive the
PMF on its state space. (Hint: Make assumptions in term of parameters whenever needed, e.g., the a priori
probability governing the chance that the coin will be turning out a head.)
What’s the expected value of dollars you will win?
Solution.
Let Assume , , define random variable , then the PMF on as is Hence 2. The Random Experiment changes to “Tossing a coin (not necessarily a fair one) times; if the coin turns out a
head, then you earn 1 dollar, otherwise, 0 dollar.” You are still interested in how many dollars you will win after
finishing the experiment, i.e., tossing times. Define a new random variable to describe the experiment whose state
space should be
and derive the PMF on its state space. Assume
.
What’s the expected value of dollars you will win? What’s the probability that you will win at least some money?
Solution.
Let , . Define random variable as Assume , the PMF on is
() where we have written () as . The main reasoning involved in getting the above result is to observe:
(i) there are totally ( ) outcomes that has heads and tails; (ii) for each such outcome the probability for observing it is . Hence ∑ () ∑ ∑( ∑ ) ∑( ) † and 3. The Random Experiment changes again, to “Tossing a coin (not necessarily a fair one) indefinitely until it turns out
a head. The number of dollars you will win is equal to the number of tosses it takes to see the first head turning out.”
You are still interested in how many dollars you will win after finishing the experiment, which equals the number of
tosses until you see the first time a head turns out. Define a new random variable to describe the experiment whose
state space should be
and derive the PMF on its state space. Assume
.
What’s the expected value of dollars you will win?
What’s the probability that you will win at least dollars? Solution.
Let
as Assume
† , then , then the PMF An easier way is to treat the
. is a countably infinite set. on . Define random variable is defined here as the sum of of the previous question and exploit the linearity of : where we have written as . Hence ∑ ∑ ∑ ∑ ( ) and 4. The Random Experiment changes yet again, to “Tossing a coin (not necessarily a fair one) indefinitely until heads
have been observed. The number of dollars you will win is equal to the total number of tosses until you see the th
head turning out.” You are still interested in how many dollars you will win after finishing the experiment, which
equals the number of tosses until you see the th head. Define a new random variable to describe the experiment
whose state space should be
and derive the PMF on its state space. Make necessary
parametric assumptions when needed. Assume
.
What’s the expected value of dollars you will win?
Solution.
Let , . Define random variable as Assume , then the PMF on is
( where we have written ) as . We now proceed to compute the expected value. First we extract an identity from the fact that ∑( ) ∑ ∑ ∑ ∑ hence ∑ is a PMF: A simpler way of deriving the expected value is to start with viewing a negative binomial r.v. as the sum of
geometric r.v.’s.
5. (a) Verify that Poisson distribution can approximate Binomial distribution when the number of Bernoulli trials is
very large and
is very small, while the mean remain finite. To be precise, suppose has a binomial
distribution with parameters and . If
and
as
then (b) , , , and . Compare the values of and . Solution.
(a) The following manipulation is purely technical: as ,
( (
Now it remains to prove that → )( ) ( ) )
. This is because
( )( ) ( )→ and q.e.d.
(b) This part gives you the practical explanation of the term “Poisson approximation to Binomial”. It also shows how
possibly tedious can a statistical problem be and by this way motivates you to use software packages such as the
free R language (cran.rproject.org, type ?dbinom and ?dpois with the question mark to see help files on how to
use these builtin pdf functions, if you decide to give it a try). ( ) ( ) ( )
( ) 6. (a) Show that ∫ ( ) and find the integration. (b) Show that ∫ and find the integration. (c) Show that , , and hence for : integer. (d) Find the normalizing constant such that the function is a valid PDF.
Solution.
(a)
() ∫ ∫ ( ∫ Let ∫ () ⁄ ∫ ) () , then
⁄ ∫ ∫∫ ∫ ∫ ∫ √
(b)
∫
(c)
∫ ∫ [ ∫ ∫ Since and for each integer
, we will have
if we have
th
. Then, by principle of induction (the 5 Peano’s axiom for natural numbers), we can have the
induction hypothesis
for real.
(d) Since the integration of ∫ over the entire real axis must be 1, we shall have ∫ √ √ ∫ √ √ √ 7. (a) Write down the PDF of , derive the expectation and variance.
( (b) Write down the PDF of ), its expectation and variance. (c) Write down the CDF of , its expectation and variance. (d) Derive the MGF of . (Hint: Solution.
(a) The pdf is , . The trick of remembering this is to observe the resemblance of the kernel to the function and to understand how the additional scaling parameter is
incorporated.
∫ ∫ ∫ ∫
[ (b) The pdf is the just the ( ) pdf: () (c) The pdf is just the pdf: Thus
∫ ∫ (d)
∫ √ √ ∫ √ ∫ ...
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.
 Spring '11
 Dr.Yun

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