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SADC Course in Statistics Joint distributions (Session 05)

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To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session you will be able to describe what is meant by a joint probability density function explain how marginal conditional probability distribution functions can be derived from the joint density function compute joint and marginal probabilities corresponding to a two-way frequency table

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To put your footer here go to View > Header and Footer 3 Bivariate distributions In many applications one has to work with two or more random variables at the same time. To determine the health of a child one needs to consider the age, weight, height and other variables. A function f is a bivariate joint probability mass /density function if

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To put your footer here go to View > Header and Footer 4 Example 1 Consider a trial where a coin and a die are tossed. How many outcomes are possible? Table below shows the possibilities. We will return to this table shortly. Die outcomes Coin outcomes123456 H T

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To put your footer here go to View > Header and Footer 5 Marginal Distributions Given the bivariate joint probability mass/ density function f, the marginal mass/ densities f X and f Y are defined as: The sums are for the discrete cases while the integrals are for the continuous cases.

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To put your footer here go to View > Header and Footer 6 The conditional probability mass/density function of X given Y = y is defined as Notice that the above definition resembles very closely to the definition of conditional probability. Conditional Distributions

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To put your footer here go to View > Header and Footer 7 Independent random variables Random variables X and Y are said to be independent if and only if that is, the joint mass/density function is equal to the product of the marginal mass/density functions. It follows that if X and Y are independent, then

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To put your footer here go to View > Header and Footer 8 Back to Example 1 Outcomes of die (X) f Y (y) Coin outcomes(Y)123456 H1/2 T f X (x) 1/6 1 Note that the coin/die throwing trial corresponds to independent outcomes because what happens with the coin cannot affect the die outcome. Below are the marginal probabilities. Can you compute the joint distribution?

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To put your footer here go to View > Header and Footer 9 Example 2 (Use of condoms) 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 provided responses. The sample consisted of migrant mineworkers, sex workers and members of the local community. The following are some results for men.

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To put your footer here go to View > Header and Footer 10 Sexual behaviour and condom use Sexual Behaviour(X) Condom Use (Y) Only with regular partners Only with casual partners Total Never6174391056 Sometimes9264156 Always53133186 Total7626361398

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To put your footer here go to View > Header and Footer 11 Joint and Marginal Probabilities Sexual Behaviour(X) Condom Use (Y) Only with regular partners Only with casual partners f Y (y) Never0.4410.3140.755 Sometimes0.0660.0460.112 Always0.0380.0950.133 f X (x) 0.5450.4551.000

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To put your footer here go to View > Header and Footer 12 Class Exercise (Part I): Is condom use independent of sexual behaviour in terms of type of sexual partner? Use the definition of independence and allow for sampling errors.

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To put your footer here go to View > Header and Footer 13 Class Exercise (Part II): Calculate the conditional probability that a man from the study area has casual partners given that he always uses condoms. To do this part of the exercise, it would be helpful to first calculate conditional probabilities of X, given each value for Y. Note these down in the table below, and then answer the question above.

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To put your footer here go to View > Header and Footer 14 Conditional Probabilities of X given Y Sexual Behaviour(X) Condom Use (Y) Only with regular partners Only with casual partners Never Sometimes Always

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To put your footer here go to View > Header and Footer 15 Practical work follows to ensure learning objectives are achieved…

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