During Class

Brain Gains

1. Let $g(x) = \begin{cases}e^{-x/2} & x \geq 0\\ 0 & \text{otherwise}\end{cases}$.

  1. Find $k$ so that $f(x) = k g(x)$ is a PDF for a random variable $X$. Write down the function $f$.
  2. Write down the definite integrals that give the expected value and variance of $X$. Then use Mathematica to compute the expected value and variance of $X$.
  3. Write down the definite integrals that give $P(X\leq 1)$ and $P(X\leq 2)$. Then use Mathematica to compute these two probabilities.
  4. Write down the definite integral that gives $F(x) = P(X\leq x)$, the cumulative distribution function for $X$. Then use Mathematica to compute $F(x)$ for $x\geq 0$.
  5. What is $P(X = 2)$? Write down the definite integral that would give this probability.
  6. By hand, compute $F'(x)$ and compare it to $f(x)$.
  7. Solve $F(m) = 0.5$ for $m$. We call $m$ the 50th percentile, or the median, of $X$.
  8. Solve $F(p_{95}) = 0.95$ for $p_{95}$. We call $p_{95}$ the 95th percentile of $X$.

Group Meeting

Continuation from last time - More Practice with finding PDFs, Expected Value, Variance, and CDF

These problems are a continuation from what we did last time in groups.

2. Let $g(x) = \begin{cases}e^{-\lambda x} & x \geq 0\\ 0 & \text{otherwise}\end{cases}$ with $\lambda >0$. In Mathematica, the following code tells the computer that $\lambda$ is positive, which will simplifies the output quite a bit. 

g = Exp[-\[Lambda] x]
$Assumptions = \[Lambda] > 0
  1. Find $k$ so that $f(x) = k g(x)$ is a PDF for a random variable $X$. We call this an exponential random variable.
  2. Write down the definite integrals that give the expected value and variance of $X$. Then use Mathematica to compute the expected value and variance of $X$.
  3. Write down the definite integral that gives $F(x)$, the cumulative distribution function for $X$. Then use Mathematica to compute $F(x)$ for $x\geq 0$.
  4. By hand, compute $F'(x)$ and compare it to $f(x)$.

3. Let $g(x) = \begin{cases}1 & -3\leq x\leq 5\\ 0 & \text{otherwise}\end{cases}$.

  1. Find $k$ so that $f(x) = k g(x)$ is a PDF for a random variable $X$.
  2. Write down the definite integrals that give the expected value and variance of $X$. Then use Mathematica to compute the expected value and variance of $X$.
  3. Write down the definite integral that gives $F(x)$, the cumulative distribution function for $X$. Then use Mathematica to compute $F(x)$ for $-3\leq x\leq 5$. For $x<-3$, what is $F(x)$? For $x>5$, what is $F(x)$?
  4. By hand, compute $F'(x)$ and compare it to $f(x)$.

4. Let $g(x) = \begin{cases}1 & a\leq x\leq b\\ 0 & \text{otherwise}\end{cases}$ where $a<x<b$.

  1. Find $k$ so that $f(x) = k g(x)$ is a PDF for a random variable $X$. We call this a uniform random variable.
  2. Write down the definite integrals that give the expected value and variance of $X$. Then use Mathematica to compute the expected value and variance of $X$.
  3. Write down the definite integral that gives $F(x)$, the cumulative distribution function for $X$. Then use Mathematica to compute $F(x)$ for $a\leq x\leq b$. For $x<a$, what is $F(x)$? For $x>b$, what is $F(x)$?
  4. By hand, compute $F'(x)$ and compare it to $f(x)$.

5. Let $g(x) = \begin{cases}3 & -2 \leq x < 1\\ 5 & 1 \leq x \leq 5\\ 0 & \text{otherwise}\end{cases}$.

  1. Find $k$ so that $f(x) = k g(x)$ is a PDF for a random variable $X$.
  2. Write down the definite integrals that give the expected value and variance of $X$. Then use Mathematica to compute the expected value and variance of $X$.
  3. Compute $F(x)$.
  4. By hand, compute $F'(x)$ and compare it to $f(x)$.

Solution

We can use piecewise function notation in Mathematica. Here is an example. We need the assumptions command that $x$ is a real number to compute the CDF $F$ and actually have Mathematica do the integral. 

$Assumptions = x \[Element] Reals
g = Piecewise[{{3, -2 <= x && x < 1}, {5, 1 <= x <= 5}, {0, True}}]
Plot[g, {x, -5, 10}]
A = Integrate[g, {x, -Infinity, Infinity}]
k = 1/A
f = k g
1 == Integrate[ f, {x, -Infinity, Infinity}]
EV = Integrate[x f, {x, -Infinity, Infinity}]
Var = Integrate[(x - EV)^2 f, {x, -Infinity, Infinity}]
F = Integrate[ f, {x, -Infinity, x}]
D[F, x]

Practice: Definite Integrals and Geometric arguments

Use a geometric argument to find the value of each integral. If needed, start by drawing the region over the specified bounds. Then use Mathematica to check your solution is correct.

1. $\displaystyle\int_2^5 7 dx$

Hint

This is a rectangle. What is the base? What is the height?

2. $\displaystyle\int_0^4 f(x) dx$ where $f(x) = \begin{cases}3x & 0\leq x\leq 1 \\ 4-x & 1\leq x \leq 4\end{cases}$

Hint

This is a triangle. What is the base? What is the height?

3. $\displaystyle\int_0^4 \sqrt{16-x^2} dx$

Hint

This is one fourth of a circle centered at (0,0) with radius 4. Note that $y=\sqrt{16-x^2}$ means $y^2 = 16-x^2$ or $x^2+y^2=16$. You'll encounter functions like this one when working on Knewton Alta homework.

4. $\displaystyle\int_0^{10} \sqrt{25-(x-5)^2} dx$

Hint

This is one half of a circle centered at (5,0) with radius 5. Use software to draw the region if needed. You'll encounter functions like this one when working on Knewton Alta homework.

Work on Project 3 Task 2

If you finish the above early, start working on Project 3 Task 2. You'll find that you've already quite a bit of that project as you worked together over the last few days.

Discussion

Properties of the Definite Integral

Definite Integral

For a function $f(x)$ defined on $a \leq x \leq b$, the definite integral of $f$ from $a$ to $b$ is

$\int_a^b f(x) dx = \lim_{n\to \infty }\sum_{i=1}^nf(x_i)\Delta x$,

provided the limit exists. If the limit exists, we say that $f$ is integrable on $ [ a, b ] $.

  • $f(x)$ is the integrand
  • $x$ is the variable of integration
  • $a$ is the lower bound (or lower limit of integration)
  • $b$ is the upper bound (or upper limit of integration)
  • Given $a < c < k < b$,
    $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^k f(x) dx + \int_k^b f(x) dx$.
    • Note this property also holds when the inequality $a < c < k < b$ does not hold as long as $f(x)$ is integrable (can be integrated) for all the necessary intervals.
  • $\int_a^b f(x)dx = -\int_b^a f(x)dx$.
    • One way to think about this property is that if the width for the "infinite" rectangles you are adding up are all negative for the definite integral on the right-hand side of the equation.
  • $\int_a^a f(x)dx = 0$.
    • Note this is related to the concept that $P(X = c) = 0$ for all values of $c$ when $X$ is a continuous random variable.
  • $\int_a^b cf(x)dx = c\int_a^b f(x) dx$
  • $\int_a^b f(x) + g(x) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$ AND $\int_a^b f(x) - g(x) dx = \int_a^b f(x) dx - \int_a^b g(x) dx$

Activity - Definite Integrals

Given the function $$g(x) = \begin{cases} 2 & 0 < x < 5 \\ \\ 0 & \text{otherwise} \end{cases},$$ determine whether each statement is true or false. Discuss your reasoning with your neighbor.

  • $\int_0^5 g(x)dx = 10$.
  • $\int_0^5 g(x)dx = \int_0^3 g(x)dx + \int_3^{4.5} g(x)dx + \int_{4.5}^5 g(x)dx$.
  • $\int_0^5 g(x)dx = \int_{-\infty}^0 g(x)dx + \int_0^5 g(x)dx + \int_5^{\infty} g(x)dx$.
  • $\int_0^5 g(x)dx = \int_0^3 g(x)dx + \int_1^5 g(x)dx$.
  • $\int_0^5 g(x)dx = \int_1^3 g(x)dx + \int_4^5 g(x)dx$.