Todays Goals

1
Todays Goals

• Understand what a probability density functon
  (pdf) represents.
• Calculate the Mean and Variance of Continuous
  Distributions.
• Homework 8 (due Wednesday April 1) CH379,
  110, 122, CH4 5,13, 40 plus three web problems.
• Midterm in two weeks. Wednesday April 8. Will
  cover random variables, probability models plus
  all previous material.

2
Homework 66 c

• Use total probability to calculate
• p(0) p(0R3)p(R3) p(0R6)p(R6)
• May be useful to use a tree

3
Continuous Random Variables

• Can take on any value in a specified interval
  that is, if for some A lt B, any number x between
  A and B is possible.
• P(Xx)0 for every x.
• Probability Distribution
• Definition. Let X be a continuous r.v. Then a
  probability distribution or probability density
  function (pdf) of X is a function f(x) such that
  for any two numbers a and b with a lt b,

4
Probability Density Function
is given by the area of the
shaded region.
Note It is the area under the p.d.f. curve that
has a probability interpretation- not the height
of the p.d.f. curve.

• Its graph, called the density or p.d.f. curve
  shows how the total probability of 1 is spread
  over the range of X.

5
Observe that.

• If X is a continuous r.v., then for any number
  c, P(X  c)  0. Furthermore, for any two
  numbers a and b with a lt b,

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Properties of p.d.f.

• For a function f(x) to be a legitimate p.d.f. it
  must satisfy the following properties
• f(x) gt 0 for all x
• 1.

7

• f(x) 1-x for 0 x 1
• Is f(x) a legitimate pdf?
• True for yes
• False for no

8

• f(x) -2x for -1 x 0
• Is f(x) a legitimate pdf?
• True for yes
• False for no

9

• f(x) 2-2x for 0 x 1
• What is P(X ½)?

10

• f(x) 2-2x for 0 x 1
• What is P(X lt ½)?

11

• f(x) 2-2x for 0 x 1
• What is P(X lt ½)

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Cumulative Distribution Function

• For each x, F(x) is the area under the density
  curve to the left of x.
• F(x) increases smoothly as x increases.

P(X x)
x
13
Example Uniform Distribution

• A continuous rv X is said to have a uniform
  distribution on the interval A, B if the pdf of
  X is

f(x)

• Intervals with the same size have the same
  probability associated.

1/(B-A)
x
A
B
R.v. X models a random point in the interval A,B
14
Example

• Suppose the time to complete a homework is
  uniformly distributed between 1 and 3 hours.
• What is the probability that you finish within 2
  hours?
• What is the probability that you take more than
  2.5 hours?
• What is F(x)?

15
Using F(x) to Compute Probabilities

• Let X be a continuous rv with pdf f(x) and cdf
  F(x). Then for any number a,
•  
• and for any two numbers a and b with a lt b,

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Obtaining f(x) from F(x)

• Recall that
• If X is a continuous r.v. with pdf f(x) and cdf
  F(x), then at every x at which the derivative
  F'(x) exists, F'(x)  f(x).
•  
•  
•  

17
Percentiles

• Let p be a number between 0 and 1. The (100p)th
  percentile of the distribution of a continuous
  r.v. X, denoted by ?(p), is defined by
• The median is the value of X that divides the
  area under the p.d.f. curve into to halves.
• x such that F(x)P(X lt x) 0.5.

p
h(p)
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Percentiles -- Example

• The 20th percentile of the distribution of a
  continuous r.v. X, denoted by ?(.2), is defined
  by
• The x such that F(x)P(X lt x) 0.2.

0.2
h(.2)
19
Example

• Suppose the time to complete a homework is
  uniformly distributed between 1 and 3 hours.
• What is the 95th percentile? (This means that the
  probability that you are done before this time is
  95)
• You want to be 80 sure to make an important
  date. What time should you set the date, if you
  are starting your homework at 1 pm?

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Quartiles

• The quartiles are the values that leave 25, 50
  and 75 of the distribution to the left.

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Features of a Continuous Distribution

• Formulas for ? and ?2 are the same as for
  discrete r.v. except that
• probability distribution is replaced by the
  p.d.f.,
• and summation is replaced by integration

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Example

• Calculate the expected value and variance of the
  uniform distribution on the interval A,B
• What is the expected time to complete the
  homework in the previous example?