Engineering Probability Class 15 Thurs 2018-03-08

1   Grades

2   Future Iclicker grading policy

In future classes, to get credit for a day, you'll need to get at least one question right.

3   Tutorial on probability density

Since the meaning of probability density when you transform variables is still causing problems for some people, think of changing units from English to metric. First, with one variable, X.

1. Let X be in feet and be U[0,1].

   $$f_X(x) = \begin{cases} 1& \text{if } 0\le x\le1\\ 0&\text{otherwise} \end{cases}$$

2. $P[.5\le x\le .51] = 0.01$.

3. Now change to centimeters. The transformation is $Y=30X$.

4. $$f_Y(y) = \begin{cases} 1/30 & \text{if } 0\le y\le30\\ 0&\text{otherwise} \end{cases}$$

5. Why is 1/30 reasonable?

6. First, the pdf has to integrate to 1: $$\int_{-\infty}^\infty f_Y(y) =1$$

7. Second, $$\begin{align} & P[.5\le x\le .51] \\ &= \int_.5^.51 f_X(x) dx \\& =0.01 \\& = P[15\le y\le 15.3] \\& = \int_{15}^{15.3} f_Y(y) dy \end{align}$$

4   4.8 Reliability

1. The reliability R(t) is the probability that the item is still functioning at t. R(t) = 1-F(t).

2. What is the reliability of an exponential r.v.?

3. The Mean Time to Failure (MTTF) is obvious.

4. ... for an exponential r.v.?

5. The failure rate is the probability of a widget that is still alive now dying in the next second.

6. If the failure rate is constant, the distribution is exponential.

7. The importance of getting the fundamentals (or foundations) right:

   (I mentioned this in an earlier class.)

   In the past 40 years, two major bridges in the Capital district have collapsed because of inadequate foundations. The Green Island Bridge collapsed on 3/15/77, see http://en.wikipedia.org/wiki/Green_Island_Bridge , http://cbs6albany.com/news/local/recalling-the-schoharie-bridge-collapse-30-years-later . The Thruway (I-90) bridge over Schoharie Creek collapsed on 4/5/87, killing 10 people.

   Why RPI likes the Roeblings: none of their bridges collapsed. E.g., when designing the Brooklyn Bridge, Roebling Sr knew what he didn't know. He realized that something hung on cables might sway in the wind, in a complicated way that he couldn't analyze. So he added a lot of diagonal bracing. The designers of the original Tacoma Narrows Bridge were smart enough that they didn't need this expensive margin of safety.

8. Another way to look at reliability: think of people.

   1. Your reliability R(t) is the probability that you live to age t, given that you were born alive. In the US, that's 98.7% for age 20, 96.4% for 40, 87.8% for 60.
   2. MTTF is your life expectancy at birth. In the US, that's 77.5 years.
   3. Your failure rate, r(t), is your probability of dying in the next dt, divided by dt, at different ages. E.g. for a 20-year-old, it's 0.13%/year for a male and 0.046%/year for a female http://www.ssa.gov/oact/STATS/table4c6.html . For 40-year-olds, it's 0.24% and 0.14%. For 60-year-olds, it's 1.2% and 0.7%. At 80, it's 7% and 5%. At 100, it's 37% and 32%.

9. P190: If the failure rate is constant, then the distribution is exponential. We'll show this.

10. If several subsystems are all necessary, e.g., are in serial, then their reliabilities multiply. The result is less reliable.

    If only one of them is necessary, e.g. are in parallel, then their complementary reliabilities multiply. The result is more reliable.

    An application would be different types of RAIDs. (Redundant Array of Inexpensivexxxxxxxxxxxxx Independent Disks). In one version you stripe a file over two hard drives to get increased speed, but decreased reliability. In another version you triplicate the file over three drives to get increased reliability. (You can also do a hybrid setup.)

    (David Patterson at Berkeley invented RAID (and also RISC). He intended I to mean Inexpensive. However he said that when this was commercialized, companies said that the I meant Independent.)

5   4.9 Generating r.v

Ignore. It's surprisingly hard to do right, and has been implemented in builtin routines. Use them.

6   4.10 Entropy

Ignore since it's starred.

7   Max of two r.v.

This is not in the text, but is an intro to multiple r.v.

1. pdf and cdf of the max of 2 random variables:

   If Z=max(X,Y) then \(F_Z(x) = F_X(x) F_Y(x)\)

   E.g. if X and Y and U[0,1], so \(F_X(x) = x\) for 0<=x=1, then \(F_Z(x) = x^2\)

   What are the pdf and mean here? What about the max of 3 r.v.? What about the min?

2. Iclicker. What is the cdf (for 0<=x<=1) of the max of 3 r.v. that are each U[0,1]?

   1. x
   2. x^2
   3. x^3
   4. 1
   5. 0

3. pdf of the sum of 2 r.v. If Z=X+Y then \(f_Z(z) = \int_x f_X(x) f_Y(z-x) dx\) E.g. If X and Y and U[0,1] then \(f_Z(z) =\) ?

   What is the mean?

4. The pdf of the sum of two uniform r.v. is a hat function. It looks a little more like a normal distribution than the square uniform distribution did.

   The sum of 3 uniform r.v. would look even more normal, and so on.

8   Chapter 5, Two Random Variables

1. One experiment might produce two r.v. E.g.,
   1. Shoot an arrow; it lands at (x,y).
   2. Toss two dice.
   3. Measure the height and weight of people.
   4. Measure the voltage of a signal at several times.
2. The definitions for pmf, pdf and cdf are reasonable extensions of one r.v.
3. The math is messier.
4. The two r.v. may be '''dependent''' and '''correlated'''.
5. The '''correlation coefficient''', $\rho$, is a dimensionless measure of linear dependence. $-1\le\rho\le1$.
6. $\rho$ may be 0 when the variables have a nonlinear dependent relation.
7. Integrating (or summing) out one variable gives a marginal distribution.
8. We'll do some simple examples:
   1. Toss two 4-sided dice.
   2. Toss two 4-sided ''loaded'' dice. The marginal pmfs are uniform.
   3. Pick a point uniformly in a square.
   4. Pick a point uniformly in a triangle. x and y are now dependent.
9. The big example is a 2 variable normal distribution.
   1. The pdf is messier.
   2. It looks elliptical unless $\rho$=0.

9   Material added after class

1. My handwritten tablet notes.