ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

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# Lecture 7

1. Discrete: finite or countably infinite. Contrast to continuous, to be covered later.
2. Discrete r.v.s we've seen so far:
1. uniform: M events 0...M-1 with equal probs
2. bernoulli: events: 0 w.p. q=(1-p) or 1 w.p. p
3. binomial: # heads in n bernoulli events
4. geometric: # trials until success, each trial has prob p.
3. I will attempt to play with randtool in Matlab. randtool does simulated runs for any of several probability distributions. E.g., you can simulate 100 tosses of a fair coin, and see the output histogram. Each time you rerun it, you see a different result.
See how much the results vary from run to run, but how they get more predictable as n gets larger.
4. iclicker: From a deck of cards, I draw a card, look at it, put it back and reshuffle. Then I do it again. What's the prob that exactly one of the 2 cards is a heart?
• A: 2/13
• B: 3/16
• C: 1/4
• D: 3/8
• E: 1/2
5. iclicker: From a deck of cards, I draw a card, look at it, put it back and reshuffle. I keep repeating this. What's the prob that the 2nd card is the 1st time I see hearts?
• A: 2/13
• B: 3/16
• C: 1/4
• D: 3/8
• E: 1/2
6. 3.3.2 page 109 Variance of an r.v.
1. how wide is its distribution
2. {$\sigma^2_X = VAR[X] = E[(X-m_X)^2] = \sum (x-m_x)^2 p_X(x)$}
3. standard deviation {$\sigma_X = \sqrt{VAR[X]}$}
4. {$VAR[X] = E[X^2] - m_X^2$}
5. 2nd moment: {$E[X^2]$}
6. also 3rd, 4th... moments, like a Taylor series for probability
7. shifting the distribution: VAR[X+c] = VAR[c]
8. scaling: {$VAR[cX] = c^2 VAR[X]$}
7. Example 3.20 3 coin tosses
1. general rule for binomial: VAR[X}=npq
8. iclicker: The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating for a total of 100 times. The random variable is the # of hearts seen, from 0 to 100. What's the mean of this r.v.?
• A: 1/4
• B: 25
• C: 1/2
• D: 50
• E: 1
9. iclicker: The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating for a total of 100 times. The random variable is the # of hearts seen, from 0 to 100. What's the variance of this r.v.?
• A: 3/16
• B: 1
• C: 25/4
• D: 75/4
• E: 100
10. Example 3.21 Variance of Bernoulli r.v.
11. Example 3.22 Variance of geometric r.v.
12. iclicker: The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating until you see a heart. The random variable is the # of cards you draw until that happens. What is the mean?
• A: 4
• B: 4/3
• C: 1/4
• D: 12
• E: 52
13. iclicker: The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating until you see a heart. The random variable is the # of cards you draw until that happens. What is the variance?
• A: 4
• B: 4/3
• C: 1/4
• D: 12
• E: 52
14. 3.4 page 111 Conditional pmf
15. Example 3.23 random clock - skip
16. Example 3.24 Residual waiting time
1. X, time to xmit message, is uniform in 1...L.
2. If X is over m, what's prob that remaining time is j?
3. {$p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)$}
17. {$p_X(x) = \sum p_X(x|B_i) P[B_i]$}
18. Example 3.25 p 113 device lifetimes
1. 2 classes of devices, geometric lifetimes.
2. Type 1, prob {$\alpha$}, parameter r. Type 2 parameter s.
3. What's pmf of the total set of devices?