Question

What is the solution to this Probability Density Function and how did they get 5/8? How to find the area?

Answer 1

Answer:

4.1.1 Probability Density Function (PDF)

To determine the distribution of a discrete random variable we can either provide its PMF or CDF. For continuous random variables, the CDF is well-defined so we can provide the CDF. However, the PMF does not work for continuous random variables, because for a continuous random variable P(X=x)=0 for all x∈R. Instead, we can usually define the probability density function (PDF). The PDF is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length. To get a feeling for PDF, consider a continuous random variable X and define the function fX(x) as follows (wherever the limit exists):

fX(x)=limΔ→0+P(x<X≤x+Δ)Δ.

The function fX(x) gives us the probability density at point x. It is the limit of the probability of the interval (x,x+Δ] divided by the length of the interval as the length of the interval goes to 0. Remember that

P(x<X≤x+Δ)=FX(x+Δ)−FX(x).

So, we conclude that

fX(x)=limΔ→0FX(x+Δ)−FX(x)Δ

=dFX(x)dx=F′X(x),if FX(x) is differentiable at x.

Thus, we have the following definition for the PDF of continuous random variables:

Step-by-step explanation:

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