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

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# Lecture 7

- Discrete: finite or countably infinite. Contrast to continuous, to be covered later.
- 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 prob p.

- 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. - 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

- 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

- 3.3.2 page 109 Variance of an r.v.
- how wide is its distribution
- {$ \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] $}
- also 3rd, 4th... moments, like a Taylor series for probability
- shifting the distribution: VAR[X+c] = VAR[c]
- scaling: {$ VAR[cX] = c^2 VAR[X] $}

- Example 3.20 3 coin tosses
- general rule for binomial: VAR[X}=npq

- 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

- 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

- Example 3.21 Variance of Bernoulli r.v.
- Example 3.22 Variance of geometric r.v.
- 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

- 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

- 3.4 page 111 Conditional pmf
- Example 3.23 random clock - skip
- Example 3.24 Residual waiting time
- X, time to xmit message, is uniform in 1...L.
- If X is over m, what's prob 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] $}
- Example 3.25 p 113 device lifetimes
- 2 classes of devices, geometric lifetimes.
- Type 1, prob {$\alpha$}, parameter r. Type 2 parameter s.
- What's pmf of the total set of devices?