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Section 8.5 : Probability
3. Determine the value of c for which the function below will be a probability density function.
f(x)={c(8x3−x4)if 0≤x≤80otherwise Show SolutionThis problem is actually easier than it might look like at first glance.
First, in order for the function to be a probability density function we know that the function must be positive or zero for all x. We can see that for 0≤x≤8 we have 8x3−x4≥0. Therefore, we need to require that whatever c is it must be a positive number.
To find c we’ll use the fact that we must also have ∫∞−∞f(x)dx=1. So, let’s compute this integral (with the c in the function) and see what we get.
∫∞−∞f(x)dx=∫80c(8x3−x4)dx=c(2x4−15x5)∣∣∣80=81925cSo, we can see that in order for this integral to have a value of 1 (as it must in order for the function to be a probability density function) we must have,
c=58192