# Activity 10/22/2021 ## STAT 445 / 645 -- Section 1001
Instructor: Colin Grudzien
## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ## Activities: To perform some integration techniques, we will want the following packages, {r} require(cubature) require(caTools)  ### Activity 1: Integration of density functions. Let $X$ be a random variable distributed according to a CDF $F$, defining the probability distribution. If $f$ is a continuous probability density function for that distribution, then $$\int_{-\infty}^x f(t)\mathrm{d}t = F(x) = P(X \leq x)$$ We will now show how this relationship can be computed in R with the integrate command. #### Question 1: Recalling the form of the density function and CDF for the normal, find out by how much the integrate function applied to the density differs from the built-in CDF function for calculating $P(X \leq 1)$ for $X\sim N(0,1)$. Call the abs.error variable to find the estimated error of the integral. Is the estimate too high or too low in this example? #### Question 2: Recalculate the above, but set the rel.tol keyword in the integrate function to 10e-10. Now what is the result of the absolute difference with the pnorm calculation? How does the error estimate look in relationship to the difference with the pnorm value? #### Question 3: We now change the standard normal to the normal with $\mu=-10$ and $\sigma=5$. You must pass these arguments as keyword arguments in the integrate function, using the same keywords from the dnorm function. The integrate will pass these arguments to the integrand function if specified correctly. Try computing the integral of the density function for $N(-10,5)$ to compute $P(X\leq 1)$ using the integrate function with the default rel.tol argument and the rel.tol=10e-10. How do these approximate integrals compare to the value from the pnorm function?