Math 201 Chapter 4: Probability

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Math 201 Chapter 4 Probability

• 4.3 Random Variables
• 4.4 Means Variances of RVs

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Last Time Probability Models and RVs

• ME, independent
• Multiplication rule for indept events
• Addition rule for disjoint events
• Tree diagrams
• RVs (Discrete, Continuous)
• Probability Distributions
• Probability histograms

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4.3 Random Variables (RV)

• Let X of Heads among three tosses
• Random Variable
• assigns a to each outcome in the sample space
• S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
• 3, 2, 2, 1, 2, 1, 1,
  0
• S 0, 1, 2, 3

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Probability Distribution of X

• Probability Distribution
• Value of X 0 1 2 3
• prob 1/8 3/8 3/8 1/8

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Finding Probabilities

• Probability Distribution
• Value of X 0 1 2 3
• Prob 1/8 3/8 3/8 1/8

• Toss coin 3 times
• X of heads among the two tosses.
• P(X2)
• P(at most one head)
• P(at least one head)

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Continuous R.V.S

• Example
• Any random number between 0 and 1.
• Continuous
• S all numbers x such that 0 lt x lt 1
• Can't list them all
• How can we find prob.s?

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Strategy

• Use the density curve
• Recall area under density curve is 1
• Prob.s are equal to the corresponding area under
  the curve

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Random Numbers Example 4.52
Find the probability of getting a random number
that is greater than 0.3? Area of the blue
rectangle!
Ht?
Page 317
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Normal Probability Models

• Convert values of the endpoints of the interval
  of interest to standardized scores (z scores),
• Find probabilities from Table A.
• Example page 318 56

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• 4.4 Means and Variances of RVs

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Mean of a RV

• A box contains
• 9? 1 bills
• 1? 100 bill
• Draw one bill and you win what you drew.
• X pay off
• Prob. Distribution of X?
• Long-run average won?

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Difference b/w x and mx
Played n4 times Average x103/4 25.75
Play infinitely many times Average m10.90
1
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Law of Large No.s (LLN)
As the No. of observations drawn increases, the
sample mean x approaches population mean m
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In General

• Probability Distribution
• Value of X x1 x2 x3 . xn
• probability p1 p2 p3 . pn

• The Mean (expected value) of a discrete rv is
• mX E(X) S xi pi
• This is the long-term average value that would
  result if the random process were repeated over
  and over.

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What Else Consider?

• Similar to distributions of data, also consider
  the spread or variability in the values.
• Did you hear about the guy whose hair was on fire
  and his feet were frostbitten?
• On average he was fine ?

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Variance and Standard deviation

• The variance of a rv is
• s2 S(xi-m)2 pi
• The standard deviation is
• the square root of s2
• It measures the spread (Average deviation from
  m)

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Standard deviation

• Previous Example
• Recall m10.90
• s2 S(xi-m)2 pi
• s2 1- 10.90 2(.9)100- 10.90 2(.1)
• 882.09 dollars2
• s 29.70

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Try it!

• Probability Distribution of Score
• Score 0 1 2 3
  4
• prob 0.07 0.09 0.34 0.32 0.18

Average of scores? Standard deviation of scores
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• 18 red numbers
• 18 black numbers
• 2 green (0 and 00)
• If you bet 1 on a color (Red or Black) and win,
  you get 2
• net gain 1 if you win
• -1 if you lose
• Let X net winnings if bet on a color.
• Determine the probability distribution of X

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Example Roulette

• Let Xnet winnings if bet color
• x -1 1
• p(x) 20/38 18/38

E(X) -0.0526 lose 5 cents on average
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How Do Casinos Make Money?

• If only winning 5 cents each play, on average
• Law of Large Numbers Over many, many
  observations, the average value will be very
  close to the expected value.

1 game ? lost 5 cents on ave ?Total loss 5
cents 10 game ? lost 5 cents on ave ? Total loss
50 cents 1000 game ? lost 5 cents on ave ?Ave
Total loss 5000 cents
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Caution

• Its important to be able to distinguish between
• a distribution of data and a probability
  distribution
• sample mean vs. m
• sample variance s2 vs. s2

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Homework

• 4.49, 4.51, 4.53, 4.55
• 4.115
• Quiz Friday on Chapter 4 (4.1-4.4)