How Do You Calculate the PDF for Mixed Weather-Dependent Demand Distributions?

[singhr]

Homework Statement

demand for raincoats
demand X is uniform [0,a) on a sunny day and ~U[0,b) on a rainy day. its is stated that b>a. the probability it is a sunny day is 'p' and that it is a rainy day is '1-p' (p and 1-p are constants)
compute the pdf?

Homework Equations

f(x) = f(x/B)h(B)+f(x/G)h(G)

The Attempt at a Solution

so i get a overall demand pdf f(x) = (p/a)+((1-p)/b) 0<x<a

((1-p)/b) a<x<b

Is this correct? does the demand distribution on a rainy day change when we restrict it between 0 to a

 

[mighty2000]

as well?

Your attempt at a solution is on the right track, but there are a few things that need to be corrected.

Firstly, the demand for raincoats can only be between 0 and a on a sunny day, and between 0 and b on a rainy day. This means that the PDF should be 0 for any values of x that fall outside of these ranges.

Secondly, the overall demand PDF should take into account both the sunny and rainy day scenarios. This means that the PDF should be a combination of the PDF for a sunny day and the PDF for a rainy day, weighted by their respective probabilities (p and 1-p).

So the correct PDF would be:

f(x) = (p/a) for 0<x<a
(1-p)/b for a<x<b
0 for all other values of x