Probability Density Function

[barren parcel]

There's a wikipedia page on it, but the part you need is:

In your case, a = -infinity, and b = 1. The probability that X = 1, or X = 123, or X = 1000, is all 0, because that's the probability that X belongs to a singleton set. Thus, it doesn't matter whether we ask about Pr(a <= X <= b] or Pr(a < X < b) in this case.

Basically, sub in the bounds of integration and integrate.

no, almost

Your function is piecewise.

Draw the function f on a graph, and you just need the area between the bounds. Since it's piecewise you draw each piece individually.

That's close to what one of your two integrals would be, but your upper bound of 1 is not correct.

I'll draw the function.

Or maybe I should try to help you draw it first. What's f(0)?

[mint forge]

0 to 1

[fresh schooner]

adonhs

[barren parcel]

OK, so since it's piecewise, and functions must be well-defined on their domain, and the domain here is the real numbers, f is defined for any real number.

For 0 for example, to find f(0), see which case 0 falls in. Is it true that x >= 0 when x=0?

It's a less than or equals comparison, yes.

So, that's the case, so then f(0) uses the top case, so f(0) = 0.2e^(-0.2*0).

f(0.25) = 0.2e^(-0.2*0.25),
f(0.35) = 0.2e^(-0.2*0.35),
f(0.45) = 0.2e^(-0.2*0.45),
and so on.

What about f(-1)?

Is it true that x >= 0 when x=-1?

Correct, so we would be in the "otherwise" case, meaning f(-1) is just 0.

Similarly, f(-2) = f(-3) = f(-4) = f(-0.000000000000000001) = 0.

The subsets of the real line that make up the pieces of f are (-inf, 0) and [0, inf)

In (-inf, 0), the function is just 0, so integrating that gives nothing. In [0, inf), the function is that expression with e, so that is something, and thus you get the integral adonhs gave for the probability that X is in the set [0, 1], or the set (-inf, 1), or the set (-inf, 1]. They all have the same probability.

Integration is probably something that would be done in a course by itself, and then you'd go on to apply it in probability. Integration gets its own course in university generally.

The basic idea of integration is to find an antiderivative.

Do you remember derivatives, and rules like the chain rule?

Ok. To do your probability problem, you need to know integration, and to do this problem you need to know the chain rule, and to do that you need to know how to take the derivative of a polynomial of degree 1 and the derivative of e^x (which is easy, d(e^x)/dx = e^x). It's built up.

This is the idea of how to solve your problem: we are finding the area under the curve.

All of these topics may be a bit much to cover all at once, and can take a while to learn, so you may want to focus on them each one at a time in the order I suggested.

[orchid thistle]

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