# Probability : Density and Distribution Functions - Sketch

Assume that the proportion of commercial vehicles among users of the Humber Bridge varies randomly from day to day, with density {see attachment} over 0<x<1, where c is a constant. Show that c = 12, find the distribution function, and sketch the density and distribution functions over -1<x<2.

On what fraction of days is the proportion of commercial vehicles between 20% and 50%?

Verify that f(x) = -ln(x) for 0<x<1 is a density function and sketch it. Write A = (1/4,3/4) and B = (0,1/2). Use your density sketch to assess which is bigger, P(a) or P(A|B). Calculate both values and check your assessment.

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Solution to 3.11.

Since is a density function, by definition of a density function, we have

So,

i.e.,

i.e.,

Now we want to find the distribution function F(x). By the relation between ...

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Probability Density and Distribution Functions are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.