1st Hourly
Math 1107
Case One
Random Variables
We have a pair of dice – a fair d3 {faces 0,2,5} and
a d4 {faces 1,2,3,4} – note the probability models for the dice
below.
|
d4 |
|
|
d3 |
|
|
Face |
Probability |
|
Face |
Probability |
|
1 |
0.10 |
|
0 |
1/3 |
|
2 |
0.40 |
|
2 |
1/3 |
|
3 |
0.40 |
|
5 |
1/3 |
|
4 |
0.10 |
|
|
|
We assume that the dice operate separately and independently of each other. Suppose that our experiment consists of tossing the dice, and noting the resulting pair of faces.
1.a) List the
possible pairs, and compute a probability for each.
1.b) Define LOWTIE as either the lesser of the two faces when
unequal, or the common face value when equal. List the possible values for
LOWTIE, and compute a probability for each.
Case Two
In this
experiment we have a weird pair of dice – they are telepathically linked so
that they do not operate independently. In fact, the dice produce the following
face-pairs with the following probabilities:
|
(d2face,d4face) @ Pr{(d2face,d4face)} |
1 |
2 |
|
1 |
(1,1) @ .10 |
(2,1) @ .10 |
|
2 |
(1,2) @ 0 |
(2,2) @ .15 |
|
3 |
(1,3) @ .10 |
(2,3) @ .10 |
|
4 |
(1,4) @ .20 |
(2,4) @ 0 |
|
5 |
(1,5) @ .10 |
(2,5) @ .15 |
In
this experiment we toss this weird pair of dice and note the resulting pair of faces.
In each of the following, show your intermediate steps and work. If a rule is
specified, you must use that rule for your computation.
2.a) Compute Pr{ exactly one face shows even }
using the Additive Rule.
|
(d2face,d4face) @ Pr{(d2face,d4face)} |
1 |
2 |
|
1 |
(1,1) @ .10 |
(2,1) @ .10 |
|
2 |
(1,2) @ 0 |
(2,2) @ .15 |
|
3 |
(1,3) @ .10 |
(2,3) @ .10 |
|
4 |
(1,4) @ .20 |
(2,4) @ 0 |
|
5 |
(1,5) @ .10 |
(2,5) @ .15 |
2.b) Compute Pr{
product of the two faces > 2 } using the Complementary Rule.
|
(d2face,d4face) @ Pr{(d2face,d4face)} \ Product of Faces |
1 |
2 |
|
1 |
(1,1) @ .10 \ 1 |
(2,1) @ .10 \ 2 |
|
2 |
(1,2) @ 0 \ 2 |
(2,2) @ .15 \ 4 |
|
3 |
(1,3) @ .10 \ 3 |
(2,3) @ .10 \ 6 |
|
4 |
(1,4) @ .20 \ 4 |
(2,4) @ 0 \ 8 |
|
5 |
(1,5) @ .10 \ 5 |
(2,5) @ .15 \ 10 |
Complementary Event
Pr{ product of the two faces £ 2 } = Pr{one of (1,1), (1,2), (2,1) shows} = .1+0+.1 = .2
Pr{ product of the two faces
> 2 } = 1 - Pr{ product of the two faces £ 2 } =1 - .2 = .8
Case Three
Xyrkztin’s Syndrome
Xyrkztin's Syndrome (XS) is a rare, fictitious disease exhibiting the following symptoms:
Itchy
Skin
Progressive
Failure of Immune System Function
Progressive
Failure of Skeletal System
Perceives
Invisible, Talking Evil Frog
Involuntary
Funny Gait/Walk
Death
Upon
diagnosis of XS, treatment is initiated, and the patient is tracked over time.
Suppose further that the patients'
disease
status is classified as: Fatal, Severe, Moderate, Mild, Cure/Remission, and
that the table below gives the
probabilities
for the population of XS patients at 5 years past diagnosis:
|
Status 5 Years
after Diagnosis |
Probability |
|
Fatal |
.35 |
|
Severe |
.10 |
|
Moderate |
.15 |
|
Mild |
.25 |
|
Remission/Cure |
.15 |
|
Total |
1.00 |
In our experiment, we draw individual patients (with replacement) from the XS patient population, noting the severity of the case.
3.a) Interpret the probabilities in terms of
repeated trials of draws with replacement from the XS patient (at 5 years past
diagnosis) population.
Repeated
draws with replacement from the population of
XS patients, five years past diagnosis, will yield approximately 35% of
sampled cases as fatal, 10% of sampled cases as severe, 15% of sampled cases as
moderate, 25% of sampled cases as mild and 15% of sampled cases in remission.
3.b) Describe
the perfect sample for 75 draws with replacement from the XS patient (at 5
years past diagnosis) population. Briefly describe the relationship between
this perfect sample and actual samples of 75 draws with replacement from the XS
patient (at 5 years past diagnosis) population.
|
Status 5 Years
after Diagnosis |
Probability |
Perfect
Sample Count (n-=75) |
|
Fatal |
.35 |
.35*75
= 26.25 |
|
Severe |
.10 |
.10*75
= 7.5 |
|
Moderate |
.15 |
.15*75
= 11.25 |
|
Mild |
.25 |
.25*75
= 18.75 |
|
Remission/Cure |
.15 |
.15*75
= 11.25 |
|
Total |
1.00 |
1.00*75
= 75 |
Random samples of
n=75 XS patients (five years past diagnosis), drawn with replacement, consist of
approximately 26.25 deaths, 7.5 severe cases, 11.25 moderate cases, 18.75 mild
cases and 11.25 cured cases/cases in remission.
Case Four
Conditional Probability
Draws without Replacement: Color Bowl
Case Description:
Compute conditional probabilities for pairs of draws (without replacement).
Here is our bowl,
in tabular form:
|
Color |
# in Bowl |
Proportion of Bowl |
|
Blue |
7 |
7/15 |
|
Green |
5 |
5/15 |
|
Red |
3 |
3/15 |
|
Total |
15 |
15/15 |
Suppose that on each trial of this experiment that we make three (3)
draws without replacement from the bowl.
Compute Pr{ green shows 3rd | red shows 1st and
green shows 2nd };
|
Color |
# in Bowl |
Proportion of Bowl |
|
Blue |
7 |
7/13 |
|
Green |
4 |
4/13 |
|
Red |
2 |
2/13 |
|
Total |
13 |
13/13 |
Pr{ green shows 3rd
| red shows 1st and green shows 2nd } = 4/13 @ .3077
Compute Pr{ red shows 3rd
| red shows 1st and red shows 2nd };
|
Color |
# in Bowl |
Proportion of Bowl |
|
Blue |
7 |
7/13 |
|
Green |
5 |
5/13 |
|
Red |
1 |
3/13 |
|
Total |
13 |
15/15 |
Pr{ red shows 3rd | red shows 1st and red shows 2nd
} = 1/13 @ .0769
Compute Pr{ blue shows 2nd | blue shows 1st }.
|
Color |
# in Bowl |
Proportion of Bowl |
|
Blue |
6 |
6/14 |
|
Green |
5 |
5/14 |
|
Red |
3 |
3/14 |
|
Total |
14 |
14/14 |
Pr{ blue shows 2nd | blue shows 1st }
= 6/14 @ .4286
Hint: Work these out directly - do not use the usual
formulas. Use the same approach as in Case Study 1.13.
Parkinson's disease (PD) is the second most common neurodegenerative disorder after Alzheimer's disease (AD).
Dementia is a common problem late in the course of the disease, and there is no effective therapy. Dementia severely
impairs patients' functional
status and limits the treatment of the motor manifestations of PD. No
effective therapy for
dementia in PD is
available.
Dementia is a clinical state characterized by loss of function in multiple cognitive domains.
Diagnostic features of dementia include: memory impairment and at least one of the following:
Aphasia (Diminished ability to correctly use and comprehend language. Aphasia is a language disorder caused by damage to the temporal lobe or higher up in the frontal lobe. It causes problems with receptive and expressive functions. Aphasia is an impairment in understanding and/or formulating complex, meaningful elements of language. It causes problems with words and word order making difficulties in reading and writing.)
Apraxia (A motor disorder in which voluntary movement is impaired without muscle weakness. The ability to select and sequence movements is impaired. Oral apraxia affects one ability to move the muscles of the mouth for non-speech purposes. Someone with oral apraxia would have trouble coughing, swallowing, wiggling their tongue or blowing a kiss when asked to do so. Verbal apraxia, or apraxia of speech is an impairment in the sequencing of speech sounds.),
Agnosia (An inability to recognize and identify objects or persons despite having knowledge of the characteristics of the objects or persons. People with agnosia may have difficulty recognizing the geometric features of an object or face or may be able to perceive the geometric features but not know what the object is used for or whether a face is familiar or not. Agnosia can be limited to one sensory modality such as vision or hearing. For example, a person may have difficulty in recognizing an object as a cup or identifying a sound as a cough..),
In addition, the cognitive impairments must be severe enough to cause impairment in social and occupational functioning.
The primary outcome measure in this trial is the change in patients' dementia status. The secondary outcome measures will include other scales of cognitive function, activities of daily living, mood, and quality of life.
Sketch a basic clinical trial for Donepezil in the treatment of Dementia in patients with Parkinson's Disease.
Disease of Interest: Dementia in Parkinson’s disease (PD)
Subjects of Interest: Patients with PD also presenting
dementia
Treatments of Interest: Donepezil, an anti-dementia drug,
and Placebo (matched to Donepezil)
Subject candidates diagnosed with PD who have a diagnosis of
dementia, who are briefed as to the risks and benefits
Of study participation, who are eligible for study
participation and who give informed consent (or whose proxies give informed
consent)
are randomly assigned to either Donepezil or to Placebo.
Neither study participants nor study workers are aware of
actual subject treatment assignments (double blinding).
Study subjects are tracked for modification of symptoms of
dementia: memory impairment, aphasia, agnosia, apraxia, social function,
occupational function.
Study subjects are tracked for cognitive function, quality
of life, mood.
Study subjects are tracked for drug tolerance/side
effects, as well as for toxic reactions
to drug.