Question

The governments in two countries, Statland and Statopia, are proposing to change their national currencies to the Statto. The proportion of support for such a change within the population in Statland and Statopia are modelled by random variables X and Y respectively. It is thought that the joint density function for the two variables is of the formf(x,y)=(2/5)*(2x+3y) for 0f(x,y)=0 elsewherePart b) the probability that changing to the Statto has the support of more than half the population in each country. 15/30Part c) the marginal density of the support for the Statto in Statland evaluated at the point X=1/2. is 1Part d) the marginal density of the support for the Statto in Statland evaluated at the point Y=1/2. is 1Part e) Find the conditional density of the support for the Statto in Statland given the support of a third of the population in Statopia. Evaluate this density at the point X=1/2

Answer 1

X represents the support for changing the currency in Statland, Y represents the support for changing the currency in Statopia. b) Find the probability that changing to the Statto has the support of more than half the population in each country. c) Find the marginal density of the support for the Statto in Statland at X=1/2. d) Find the marginal density of the support for the Statto in Statland at Y=1/2. e) Find the conditional density of the support for the Statto in Statland given the support of a third of the population in Statopia at X=1/2.

Step-by-step explanation:

In the given question, X represents the proportion of support for changing the currency in Statland, and Y represents the proportion of support for changing the currency in Statopia.

The joint density function for X and Y is given by f(x,y) = (2/5)(2x+3y) for 0≤x≤1, 0≤y≤1, and 0 elsewhere.

b) To find the probability that changing to the Statto has the support of more than half the population in each country, we need to find the region of the joint density function where both X and Y are greater than 0.5, and then calculate the probability in that region.

c) The marginal density of the support for the Statto in Statland evaluated at the point X=1/2 is 1.

d) The marginal density of the support for the Statto in Statland evaluated at the point Y=1/2 is 1.

e) To find the conditional density of the support for the Statto in Statland given the support of a third of the population in Statopia, we need to find the region of the joint density function where Y=1/3, and then calculate the conditional density at the point X=1/2.