MAT 325 | Probability and Statistics @ ORU
I receive an student in my office hours with questions about the area under the curve and its relationship with probability, so I decide to create a tutorial so all can benefit.
Bonus: I decide to include a second part, in how to use symbolic math to calculate integrals using R.
Here is what we worked in the board yesterday1
Here is the detailed tutorial:
In probability theory, when dealing with continuous random variables, the probability of an event occurring in a certain range corresponds to the area under the curve of the probability density function (PDF) over that range. This is because the PDF represents the likelihood of different outcomes, and integrating the PDF over a particular interval gives the cumulative probability of outcomes falling within that range.
For continuous random variables:
\[ P(a \leq X \leq b) = \int_a^b f(x) \, dx \]
Where:
Now, let’s solve two parts of an example problem using this principle.
We are given a linear probability density function:
\[ f(x) = \cases{2 - 2x& for $0 \leq x \leq 1$\\ 0,& elsewhere} \]
We will calculate the following probabilities:
This is the probability that the random variable \(x\) is less than or equal to \(\frac{1}{13}\), so we need to integrate the PDF from 0 to \(\frac{1}{13}\).
The setup is:
\[ P(x \leq \frac{1}{13}) = \int_0^{\frac{1}{13}} (2 - 2x) \, dx \]
We can break this integral into two parts:
\[ P(x \leq \frac{1}{13}) = \int_0^{\frac{1}{13}} 2 \, dx - \int_0^{\frac{1}{13}} 2x \, dx \]
Let’s calculate each term:
\[ \int_0^{\frac{1}{13}} 2 \, dx = 2x \Big|_0^{\frac{1}{13}} = 2 \times \frac{1}{13} - 2 \times 0 = \frac{2}{13} \]
For the second term:
\[ \int_0^{\frac{1}{13}} 2x \, dx = x^2 \Big|_0^{\frac{1}{13}} = \left( \frac{1}{13} \right)^2 - 0^2 = \frac{1}{169} \]
Now subtract the two results:
\[ P(x \leq \frac{1}{13}) = \frac{2}{13} - \frac{1}{169} \]
To simplify, find a common denominator:
\[ \frac{2}{13} = \frac{26}{169} \]
So:
\[ P(x \leq \frac{1}{13}) = \frac{26}{169} - \frac{1}{169} = \frac{25}{169} \]
Thus, the probability is:
\[ P(x \leq \frac{1}{13}) = \frac{25}{169} \]
For this part, we need to find the probability that \(x\) is greater than 0.77, which means we will integrate the PDF from \(0.77\) to 1:
\[ P(x > 0.77) = \int_{0.77}^{1} (2 - 2x) \, dx \]
As before, break this into two integrals:
\[ P(x > 0.77) = \int_{0.77}^{1} 2 \, dx - \int_{0.77}^{1} 2x \, dx \]
The first integral is straightforward:
\[ \int_{0.77}^{1} 2 \, dx = 2x \Big|_{0.77}^{1} = 2 \times 1 - 2 \times 0.77 = 2 - 1.54 = 0.46 \]
For the second term:
\[ \int_{0.77}^{1} 2x \, dx = x^2 \Big|_{0.77}^{1} = 1^2 - 0.77^2 = 1 - 0.5929 = 0.4071 \]
Now subtract the two results:
\[ P(x > 0.77) = 0.46 - 0.4071 = 0.0529 \]
Thus, the probability is:
\[ P(x > 0.77) = 0.0529 \]
To calculate probabilities for continuous random variables, we integrate the probability density function over the desired range, which gives us the area under the curve. In this example:
When dealing with continuous random variables, probabilities correspond to the area under the curve of the probability density function (PDF) over a given range. Symbolic mathematics can simplify this process by allowing you to perform exact calculations in R.
In this tutorial, we will use the Ryacas package to compute these integrals symbolically.
First, install and load the Ryacas package, which allows you to perform symbolic computations in R, such as symbolic integration.
install.packages("Ryacas")
library(Ryacas)For this example, we have the PDF \(f(x) = 2 - 2x\), which is valid for \(0 \leq x \leq 1\). We’ll define this PDF symbolically using the ysym() function.
f <- ysym("2 - 2*x") # Define the PDF symbolicallyTo find the probability that \(x\) is less than or equal to \(\frac{1}{13}\), we integrate the PDF from \(0\) to \(\frac{1}{13}\) symbolically.
Pa <- integrate(f, "x", 0, 1/13) # Calculate the integral for P(x <= 1/13)
print(paste("P(x <= 1/13) =", Pa))Here:
integrate() performs the symbolic integration of the function f with respect to \(x\) over the interval from \(0\) to \(\frac{1}{13}\).Pa and then printed.To compute the probability that \(x\) is greater than \(0.77\), we integrate the PDF from \(0.77\) to \(1\):
Pb <- integrate(f, "x", 0.77, 1) # Calculate the integral for P(x > 0.77)
print(paste("P(x > 0.77) =", Pb))This integrates the PDF from \(0.77\) to \(1\) and stores the result in Pb.
The results from the symbolic integration give the exact probabilities. The paste() function is used to display the result with a clear message.
#install.packages("Ryacas")
library(Ryacas)
# Define the PDF symbolically
f <- ysym("2 - 2*x")
# Calculate P(x <= 1/13)
Pa <- integrate(f, "x", 0, 1/13)
print(paste("P(x <= 1/13) =", Pa))[1] "P(x <= 1/13) = 25/169"# Calculate P(x > 0.77)
Pb <- integrate(f, "x", 0.77, 1)
print(paste("P(x > 0.77) =", Pb))[1] "P(x > 0.77) = 0.0529"This tutorial demonstrated how to set up and solve integrals to calculate probabilities by finding the area under the curve, showing how this area corresponds to the likelihood of outcomes within specified ranges. Additionally, we explored how to perform the same calculations symbolically using the Ryacas library in R, providing a powerful tool for solving integrals with exact precision and gaining deeper insight into the underlying mathematics.
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