Uniform

1
Uniform

• Continuous RV are defined over an interval
  (possibly infinite)
• Described by a PDF prob density function
• Prob(altXltb) INT f(x) from a to b

2
Uniform

• For a uniform RV, the pdf is a constant over
  a,b
• Since total area must be 1, the value of the pdf
  must be 1/(b-a)
• Prob(c1ltxltc2) (c2-c1)/(b-a)
• Assuming that c1, c2 are between a and b
• Probability of an interval is the fraction it is
  of the entire interval.

3
Uniform

• The uniform distribution is a no information
  model
• Puts no preference for one value over another
• DOES mean that we know the max possible range
  a,b

4
Uniform

• Could model the amount of change for a bill
• Unif(0,99)
• Could model location of bits of litter along the
  hallway as uniform over the length of the hallway

5
Uniform

• For Unif(a,b), mean(ab)/2
• SD (b-a)/v12
• Easiest to derive for Unif(0,1)
• Then shift and scale to get Unif(a,b)

6
Uniform

• Exercises p. 187, 4.1.4-4.1.6

7
Uniform

• Simulation
• Rand(m,n) returns a mxn matrix of unif(0,1)
• (rand(1,10)lt.3) returns 10 Bernoulli RVs with
  p0.3

8
Uniform

• Pseudorandom numbers
• rand('state',0) returns to initial state
• rand('state',081234) generates diff set of random
  s

9
Uniform

• Horserace game
• Rows labeled 2-12
• 6 columns
• Roll 2 dice and advance the horse whose number
  came up
• First horse to reach col 6 wins
• What is probability horse 7 wins?
• Horse 8?

10
Uniform

• floor(6rand(1,10)1) will simulate one die
• See horse.m to simulate horse race

11
Uniform

• With 10,000 trials

12
Uniform

• Results are not exactly the same
• Will see later how to put confidence bounds on
  the results
• Note If need to advance 10 squares to win,
  Prob(7 wins) 0.42
• So answer depends on how long the race is

13
Uniform

• Inference
• Consider Unif(0,A) so we only have 1 parameter to
  deal with
• A is the theoretical max value
• Suggests we use the max of our data to estimate
  (or test) A
• Max is an order statistic