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:
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.
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.
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:
\[\int_{-\infty}^\infty x f(x) \, dx.\]
This comes from the fact that the average \(x\) value of the region under any nonnegative function is:
\[\bar{x} = \frac{\int_a^b x f(x) \, dx}{\int_a^b f(x) \, dx} = \frac{1}{\text{Area}} \int_a^b x f(x) \, dx\]
Here is a pretty good video explanation of what we covered today.
Today we talked about regions between two curves. We looked at the following topics:
\[ \text{Area} = \int_a^b [f(x)-g(x)] \, dx.\]
\[ \bar{x} = \frac{1}{\text{Area}} \int_a^b x [f(x)-g(x)] \, dx.\]
\[ V = 2\pi \bar{x} A.\]
We did the following examples in class:
Find the area enclosed by the parabolas \(y = x^2\) and \(y = 2x-x^2\).
Find the area between \(y = e^x\) and \(y = x\) that is bounded by \(x = 0\) and \(x=1\).
Suppose two cars start driving in the same direction at time \(t=0\). Let \(v = f(t)\) represent the velocity of car A and \(v=g(t)\) represent the velocity of car B. What does the area between the curves \(v = f(t)\) and \(v=g(t)\) represent about the cars?
Find the area enclosed by the line \(y = x-1\) and the parabola \(y^2 = 2x+6\).
Find the average \(x\) and \(y\) values of the triangular region between the line \(y = 4-2x\) and the x and y-axes.
Show that the function \(f(x) = 12x^2 - 12x^3\) is a probability density function on the interval \([0,1]\). Then find the theoretical average value of random variable with this PDF.
Use the Theorem of Pappus to find the volume of the donut shaped solid obtained by revolving the circle of radius 1 centered at \((2,0)\) around the \(y\)-axis.
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:
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?
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 kinectic energy is \(\frac{1}{2}mv^2\).)
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?
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: \[dF =(\text{Volume})(\text{Weight Density}) dx\] and the weight density of water is 9800 Newtons per meter cubed.
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.)