Math 142 - Week 8 Notes

Monday, April 12

Today we talked about how integrals apply to probability theory. We introduced probability distribution functions which are functions that are always nonnegative and have an integral that equals one. They are used to model continuous random variables.

We focused on three special types of probability distributions:

All of these examples model a random experiment where the outcome \(x\) is unknown.

Major concept: The probability that \(x\) ends up between \(a\) and \(b\) is the integral of the probability density function: \[P(a \le x \le b) = \int_a^b f(x) \, dx.\]

We did the following examples:

  1. Suppose that x is a random number with a uniform distribution between \(a=2\) and \(b=5\), that is the probability density function (PDF) for \(x\) is \[f(x) = \begin{cases} \frac{1}{3} & \text{ if } 2 < x < 5 \\ 0 & \text{ otherwise.} \end{cases}\] Find the probability that \(x\) is 4 or more.

  2. Large asteroids with diameter greater than 1 kilometer hit the Earth every few million years. The time \(x\) (measured in millions of years) until the next large asteroid hits the Earth is estimated to have the exponential distribution with PDF: \[f(x) = 2 e^{-2x}, ~ x > 0.\] Find the probability that a large asteroid will hit the Earth in the next 1 million years.

  3. If \(x\) has a standard normal distribution, then the PDF is \[\frac{1}{\sqrt{2\pi}} e^{-x^2/2}\] which is too hard to integrate directly. Instead, use a Riemann sum to estimate the probability that \(x\) is between \(+1\) and \(-1\).

We also talked about how to find the theoretical average outcome (also known as the expected value) by integrating:

\[\int_{-\infty}^\infty x f(x) \, dx.\]

  1. Find the expected value of a random number between 2 and 5 if it has the uniform distribution on \([2,5]\).

  2. Use the probability density function \(f(x) = 2 e^{-2x}\) to find the theoretical average length of time until the next large asteroid hits the Earth.

  3. If \(x\) has the standard normal distribution, then \(|x|\) has the probability distribution \(\displaystyle f(x) = \begin{cases} \frac{2}{\sqrt{2\pi}} e^{-\tfrac{1}{2}x^2} & \text{ if } x \ge 0 \\ 0 & \text{ otherwise.} \end{cases}\). Find the expected value of a random value with this PDF.

  4. Let \(x\) be a random number between 0 and 1 that has a linear PDF function as shown below.

    1. Find a formula for the PDF function \(f(x)\).

    2. Find the probability that \(x < \frac{1}{2}\).

    3. Find the expected value of \(x\).

1

Here is a pretty good video explanation of what we covered today.


Wednesday, April 14

Today we talked about three things:

  1. How to find the average x-value of a region between two curves.

\[ \bar{x} = \frac{1}{A} \int_a^b x [f(x)-g(x)] \, dx.\]

  1. How to find the average y-value of a region between two curves.

\[ \bar{y} = \frac{1}{2A} \int_a^b f^2(x)-g^2(x) \, dx.\]

  1. Theorem of Pappus. The volume of the solid formed by revolving a region around a line is \(V = 2 \pi r A\) where \(A\) is the volume of the region and \(r\) is the distance between the line and the centroid of the region.

The centroid is the point with coordinates \((\bar{x},\bar{y})\).

We did the following examples in class:

  1. Find the average x-values of the triangular region between the line \(y = 4-2x\) and the x and y-axes.

  2. Find the average x-value of the quarter circle under \(y = \sqrt{16-x^2}\) from \(x=0\) to \(x=4\).

  3. Find the average x-value of the region under \(f(x) = e^{-x}\) from \(x =0\) to \(\infty\). This function is an example of a PDF (probability density function). For the region under a PDF, the expected value is the same as the average x-value (since the area under a PDF is always 1).

  4. Find the average y-value of the region between the parabola \(y = 4-x^2\) and the line \(y = 3\).

  5. Use the Theorem of Pappus to find the volume of the donut shaped solid obtained by revolving the circle of radius 1 centered at \((0,2)\) around the \(x\)-axis.


Friday, April 16

Today we talked about work and energy. Here is a longish video that explains most of what we talked about in class today. For a variable force \(F= F(x)\), work is the integral of force with respect to distance: \[W = \int F \, dx.\]

We did the following examples:

  1. A spring has force \(F = 10 x\) where \(x\) is the amount the spring is compressed (in meters) and the force is in Newtons. Find the work needed to compress the spring 1 meter?

  2. If the compressed spring in the last problem pushes a 1 kg object along a slippery surface so that all of the potential energy of the spring is converted to kinetic energy of the object, then how fast will the object be moving after the spring pushes it? (Hint: recall that kinetic energy is \(\frac{1}{2}mv^2\).)

  3. A ship’s anchor weighs 3000 lbs. and is attached to a chain that weighs 20 lbs. per foot. How much work is needed to raise the anchor to the surface when the water is 100 feet deep?

  4. How much work is needed to pump water into a cylindrical tank that is 10 meters above the ground and is 3 meters tall with radius 1 meter? To solve this problem use the alternative formula for work: \[W = \int x \, dF\] where \(dF\) is a the weight of a slice of water: \[dF =(\text{Weight Density})(\text{Volume})=(\text{Weight Density})(\pi r^2 \, dx) \] and the weight density of water is 9800 Newtons per meter cubed.

  5. How much work is needed to pump all of the water out of a conical pool that has a radius of 4 feet at the surface and is 6 feet deep? (The weight density of water measured in English units is 62.4 pounds per cubic foot.)