# Joint Distributions and Conditional Probabilities

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```Module H1 Practical 5
Joint Distributions and Conditional Probabilities
Getting together in pairs, discuss the following questions. After discussion some groups
will be randomly chosen to present the answers to one question to the rest of the class.
1.
A cross-sectional survey on HIV and AIDS was conducted in a major mining town
in South Africa in 2001. Among the issues investigated were sexual behaviour and
the use of condoms. A total of 2231 people between the ages of 13 to 59 were
interviewed. The sample consisted of migrant mineworkers, sex workers and
members of the local community. The following are some results for women.
Use condoms?
Sexual behaviour
Only with regular
Only with casual
partner
partners
Total
Never
Sometimes
Always
438
61
56
158
25
95
596
86
151
Total
555
278
833
(a) Calculate the joint and marginal probability distributions for the data and present it
in a tabular form.
(b) Consider whether condom use for these women are independent of sexual
behaviour (in terms of type of sexual partner). Use the definition of independence
(c) Calculate the conditional probability that a woman from the study area has casual
partners given that she always uses condoms.
Module H1 Practical 5 – Page 1
Module H1 Practical 5
2. Suppose the random variables X and Y have joint probability density function given
by
-x-y
F(x , y) = e
, 0 < x < , 0 < y < , zero elsewhere.
Find the marginal probability density functions of X and Y and show that X and Y are
independent.
3. A fair die is thrown at random three independent times. Let Y be the maximum of
these three outcomes, each outcome being the number shown on the face uppermost
on the die. Find the probability density function of Y. (Hint: Let Xi be the outcome of
the ith trial, i=1,2,3 and use the independence argument).
Module H1 Practical 5 – Page 2
Module H1 Practical 5
IF YOU HAVE TIME, TRY ALSO THE FOLLOWING:
4.
A new variety of maize has been recommended that gives higher yields but is less
resistant to stress and so farmers are concerned about changing over, fearing they may lose
out if there’s a bad year. There are two measures of interest – first of all, the yield (cob
weight) of maize harvested from each plant (in gms) and secondly the spacing of plants
used by farmers in the field. Both these show how well the maize is doing. It is known
that the average cob weight per plant of maize, x, with the old variety has probability
density function given by
f(x) = exp[–(x–)2/22]/(22)
where  = 67 gms and  = 25 gms
while the spacing, y, has the same distribution with  = 1 metre and  = 0.10 metre
(a) Write down the joint distribution for the old variety. Assume the two measures are
independent.
(b) The new variety has a joint distribution of
f ( x, y ) 
0<x<2 ;
1
2 x( 1  x )
0.6  y < 1.6
Still assuming independence, what is the marginal distribution of x?
(c) By deduction, what is the marginal distribution of y?
(d) The marginal distribution of x is called a Beta distribution. Plot this function using
Excel for a range of values of x, and consider what the implications are of having a
marginal distribution for potato yields of this shape.