Engineering Probability Class 7 Thu 2022-02-03
1 Exam one
Will be in class on Feb 28.
The test platform will be gradescope.
The questions will mostly be multiple choice and short answer.
Students with accommodations will start at 2 and run late in my lab.
If you're quarantined, tell me, and do it at home.
I cannot possibly block all the ways you might cheat. So I make a reasonable effort, and beyond that have to trust you. That's how it will work after you graduate.
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If you submit a design for, say a new airport terminal, your customer may not be able to double check your calculations. They have to assume you did it right. However:
2 Probability in the real world - enrichment
Oct. 5, 1960: The moon tricks a radar.
Where would YOU make the tradeoff between type I and type II errors?
3 Chapter 3 Discrete Random Variables
This chapter covers Discrete (finite or countably infinite) r.v.. This contrasts to continuous, to be covered later.
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Discrete r.v.s we've seen so far:
uniform: M events 0...M-1 with equal probs
bernoulli: events: 0 w.p. q=(1-p) or 1 w.p. p
binomial: # heads in n bernoulli events
geometric: # trials until success, each trial has probability p.
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3.1.1 p107 Expected value of a function of a r.v.
Z=g(X)
E[Z] = E[g(x)] = \(\sum_k g(x_k) p_X(x_k)\)
Example 3.17 p107 square law device
\(E[a g(X)+b h(X)+c] = a E[g(X)] + b E[h(x)] + c\)
Example 3.18 Square law device continued
Example 3.19 Multiplexor discards packets
Compute mean of a binomial distribution.
Compute mean of a geometric distribution.
3.3.1, page 107: Operations on means: sums, scaling, functions
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3.3.2 page 109 Variance of an r.v.
That means, how wide is its distribution?
Example: compare the performance of stocks vs bonds from year to year. The expected values (means) of the returns may not be so different. (This is debated and depends, e.g., on what period you look at). However, stocks' returns have a much larger variance than bonds.
\(\sigma^2_X = VAR[X] = E[(X-m_X)^2] = \sum (x-m_x)^2 p_X(x)\)
standard deviation \(\sigma_X = \sqrt{VAR[X]}\)
\(VAR[X] = E[X^2] - m_X^2\)
2nd moment: \(E[X^2]\)
also 3rd, 4th... moments, like a Taylor series for probability
shifting the distribution: VAR[X+c] = VAR[X]
scaling: \(VAR[cX] = c^2 VAR[X]\)
Derive variance for Bernoulli.
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Example 3.20 3 coin tosses
general rule for binomial: VAR[X]=npq
Derive it.
Note that it sums since the events are independent.
Note that variance/mean shrinks as n grows.
Geometric distribution: review mean and variance.
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Suppose that you have just sold your internet startup for $10M. You have retired and now you are trying to climb Mt Everest. You intend to keep trying until you make it. Assume that:
Each attempt has a 1/3 chance of success.
The attempts are independent; failure on one does not affect future attempts.
Each attempt costs $70K.
Review: What is your expected cost of a successful climb?
$70K.
$140K.
$210K.
$280K.
$700K.
3.4 page 111 Conditional pmf
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Example 3.24 Residual waiting time
X, time to xmit message, is uniform in 1...L.
If X is over m, what's probability that remaining time is j?
\(p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)\)
\(p_X(x) = \sum p_X(x|B_i) P[B_i]\)
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Example 3.25 p 113 device lifetimes
2 classes of devices, geometric lifetimes.
Type 1, probability \(\alpha\), parameter r. Type 2 parameter s.
What's pmf of the total set of devices?
Example 3.26, p114.
3.5 p115 More important discrete r.v
Table 3.1: We haven't seen \(G_X(z)\) yet.
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3.5.1 p 117 The Bernoulli Random Variable
We'll do mean and variance.
Example 3.28 p119 Variance of a Binomial Random Variable
Example 3.29 Redundant Systems
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3.5.3 p119 The Geometric Random Variable
It models the time between two consecutive occurrences in a sequence of independent random events. E.g., the length of a run of white bits in a scanned image (if the bits are independent).
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3.5.4 Poisson r.v.
The experiment is observing how many of a large number of rare events happen in, say, 1 minute.
E.g., how many cosmic particles hit your DRAM, how many people call to call center.
The individual events are independent. (In the real world this might be false. If a black hole occurs, you're going to get a lot of cosmic particles. If the ATM network crashes, there will be a lot of calls.)
The r.v. is the number that happen in that period.
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There is one parameter, \(\alpha\). Often this is called \(\lambda\).
\begin{equation*} p(k) = \frac{\alpha^k}{k!}e^{-\alpha} \end{equation*} Mean and std dev are both \(\alpha\).
In the real world, events might be dependent.
Example 3.32 p123 Errors in Optical Transmission
3.5.5 p124 The Uniform Random Variable