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 in forming your opinion. (c) Calculate the conditional probability that a woman from the study area has casual partners given that she always uses condoms. SADC Course in Statistics 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). SADC Course in Statistics 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–)2/22]/(22) 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. SADC Course in Statistics Module H1 Practical 5 – Page 3