How Do You Determine the Constant in a Piecewise Probability Density Function?

[someguy54]

if x is a continuous random variable from -1 to 1...how do you find c:

f(x) = c + x , -1 < x < 0
c - x, 0 < x < 1

Do I integrate each one? Where do I go from there? Thanks!

 

[HallsofIvy]

You should know that the integral of a probability density, over the entire interval, is 1: something has to happen so the probability over all possible outcomes has to be 1.
What do you get if you integrate f(x) from -1 to 1 (do as to separate parts and add them- the answer will depend on c). Set that equal to 1 and solve for c.

 

[tacman]

To find c, you will need to use the property that the total area under a probability density function (PDF) is equal to 1. This means that the integral of the PDF over the entire range of x values must equal 1.

In this case, you have two separate equations for the PDF, one for the range -1 < x < 0 and one for the range 0 < x < 1. So, you will need to integrate each of these equations separately over their respective ranges and set the sum of the two integrals equal to 1.

Let's start with the first equation, f(x) = c + x, over the range -1 < x < 0. The integral of this equation is equal to the area under the curve, which can be visualized as a triangle with base of length 1 and height of c. Using the formula for the area of a triangle, we can write the integral as:

∫f(x) dx = ∫(c + x) dx = 1/2 * (1) * c = 1/2 * c

Next, we need to do the same for the second equation, f(x) = c - x, over the range 0 < x < 1. Again, the integral represents the area under the curve, which in this case is a triangle with base of length 1 and height of c. So, the integral can be written as:

∫f(x) dx = ∫(c - x) dx = 1/2 * (1) * c = 1/2 * c

Now, to find c, we set the sum of these two integrals equal to 1:

1/2 * c + 1/2 * c = 1
c = 1/2

So, the value of c that satisfies the condition that the total area under the PDF is equal to 1 is 1/2. This means that the complete PDF for the given continuous random variable is:

f(x) = 1/2 + x, -1 < x < 0
f(x) = 1/2 - x, 0 < x < 1

I hope this helps!