Math 145

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Math 145

• February 24, 2014

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Random Variable

• A random variable is a variable whose value is
  a numerical outcome of a random phenomenon.
• A random variable is a function or a rule that
  assigns a numerical value to each possible
  outcome of a statistical experiment.
• Two Types
• 1. Discrete Random Variable A discrete random
  variable has a countable number of possible
  values (There is a gap between possible values).
• 2. Continuous Random Variable A continuous
  random variable takes all values in an interval
  of numbers.

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Examples

• Tossing a coin 3 times
• Sample Space
• HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
• Random variables
• X1 The number of heads.
• 3, 2, 2, 2, 1, 1, 1, 0
• X2 The number of tails.
• 0, 1, 1, 1, 2, 2, 2, 3

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Rolling a Pair of Dice

• Sample Space

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
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Rolling a Pair of Dice

• Random variable X3 Total no. of dots

2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
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Rolling a Pair of Dice

• X4 (positive) difference in the no. of dots

0 1 2 3 4 5
1 0 1 2 3 4
2 1 0 1 2 3
3 2 1 0 1 2
4 3 2 1 0 1
5 4 3 2 1 0
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Rolling a Pair of Dice

• X5 Higher of the two.

1 2 3 4 5 6
2 2 3 4 5 6
3 3 3 4 5 6
4 4 4 4 5 6
5 5 5 5 5 6
6 6 6 6 6 6
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More Examples

• Survey
• Random variables
• X6 Age in years.
• X7 Gender 1male, 0female.
• X8 Height.
• Medical Studies
• Random variables
• X9 Blood Pressure.
• X10 1smoker, 0non-smoker.

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Probability Distribution

• Tossing a coin 3 times
• Sample Space
• HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
• Random variable X1 The number of heads.
• 3, 2, 2, 2, 1, 1, 1, 0

x 0 1 2 3
Prob. 1/8 3/8 3/8 1/8
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Probability Histogram

• Tossing a coin 3 times
• Random variable X1 The number of heads.

X 0 1 2 3
Prob. 1/8 3/8 3/8 1/8
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Rolling a Pair of Dice

• Sample Space

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
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Rolling a Pair of Dice

• Random variable X3 Total no. of dots

2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
x 2 3 4 5 6 7 8 9 10 11 12
P 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
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Rolling a Pair of Dice

• Random variable X3 Total no. of dots

x 2 3 4 5 6 7 8 9 10 11 12
P 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
1. Pr(X3lt5) 2. Pr(3ltX3lt12)
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Discrete Random Variable

• A discrete random variable X has a countable
  number of possible values.
• The probability distribution of X

x x1 x2 x3 xk
Prob p1 p2 p3 pk

• where,
• Every pi is a between 0 and 1.
• p1 p2 pk 1.

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Mean of a Discrete R.V.

• The probability distribution of X

x x1 x2 x3 xk
Prob p1 p2 p3 pk

1. Mean (?) E(X) x1p1x2p2 xkpk
2. Variance (?2) V(X) (x1-?)2p1 (x2-?)2p2
   (xk-?) 2pk .

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Continuous Random Variable

• A continuous random variable X takes all
  values in an interval of numbers.
• Examples X11 Amount of rain in October.
• X12 Amount of milk produced by a cow.
• X13 Useful life of a bulb.
• X14 Height of college students.
• X15 Average salary of UWL faculty.
• The probability distribution of X is described
  by a density curve.
• The probability of any event is the area under
  the density curve and above the values of X that
  make up the event.

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Continuous Distributions

1. Normal Distribution
2. Uniform Distribution
3. Chi-squared Distribution
4. T-Distribution
5. F-Distribution
6. Gamma Distribution

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Thank you!