# Lecture 23 continuous random variables

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```					   Random variables can be classified as either
discrete or continuous.
   Example:
◦ Discrete: mostly counts
◦ Continuous: time, distance, etc.
   1. They are used to describe different types
of quantities.
   2. We use distinct values for discrete random
variables but continuous real numbers for
continuous random variables.
   3. Numbers between the values of discrete
random variable makes no sense, for
example, P(0)=0.5, P(1)=0.5, then P(1.5) has
no meaning at all. But that is not true for
continuous random variables.
   Both discrete and continuous random
variables have sample space.
   For discrete r.v., there may be finite or
infinite number of sample points in the
sample space.
   For continuous r.v., there are always infinitely
many sample points in the sample space.
   *** For discrete r.v., given the pmf, we can
find the probability of each sample point in
the sample space.
   *** But for continuous r.v., we DO NOT
consider the probability of each sample point
in the sample space because it is defined to
be ZERO!
   In another word,
   For discrete random variables, only the value
listed in the PMF have positive probabilities,
all other values have probability zero. We can
find probability for some specific value or an
interval of values.

   For continuous random variables, the
probability of every specific value is zero.
Probability only exists for an interval of
values for continuous r.v..
   Let X be the number of stops for a citybus
going from downtown Lafayette to Purdue
campus. X is a discrete/continuous?
   Let Y be the distance from the train station
and where a citybus can stop at when it
comes from downtown Lafayette to Purdue
campus. Y is a discrete/continuous?
   P(X=3 stops)=?
   P(Y=150 yards)=?
   PDF and CDF.
   PDF is Probability Density Function, it is
similar to the PMF for discrete random
variables, but unlike PMF, it does not tell us
   CDF is Cumulative Distribution Function, it
has a counterpart for discrete random
variables, but for continuous random
variables, it is the only way we can find a
probability.
   For discrete random variables:
◦ PMF: P(X=K)
◦ CDF: P(a < X < b) = ∑KP(X=K)
   For continuous random variables:
◦ PDF: f(x)
◦ CDF: F(x)=P(a < X < b) = ∫ab f(x)dx
   For discrete random variables, both PMF and
CDF can tell us probabilities.

   For continuous random variables, ONLY CDF
can tell us probabilities.
   Given X is a continuous random variable with
sample space Ω and its PDF is f(x), f(x) must
satisfy the following conditions:
◦ 1. 0≤ f(x)
◦ 2. ∫Ωf(x) dx= 1
◦ The same as the conditions for discrete random
variables.
   A continuous random variable X has the pdf
f(x)=c(x-1)(2-x) over the interval [1, 2] and 0
elsewhere. What value of c makes f(x) a valid
pdf for X?
   What is P(x>1.5)?
   Think about the citybus example and simplify
it. Suppose the citybus starts at point A and
goes toward point B, if this bus can stop at
will, or stop at each point between A and B
with equal probability, we let X be the
distance between where the bus stops and
point A.
   Then X is a random variable and it is said to
follow a Uniform distribution.
   We will talk about several continuous
distributions, we need to know:
◦ Their PDF
◦ How to calculate probability under those
distributions.
◦ How to find mean and variance for those random
variables
   For Uniform:
◦ PDF:
 1
       ,A X  B
f ( x)   B  A
0, elsewhere

   In order to calculate the probability, we need to
know the distance between A and B.
   In another word, the parameters for a uniform
distribution are A and B in this case, where A and
B are defined as the distance mark for the two
points.
   For example, if B is 2000 yards away from A,
then B-A=2000.
   And the probability that the bus stops within
200 yards from A would be

200                200
1         200

0
f ( x)dx    
0
2000
dx 
2000
 0.1
   Then what is the probability that the bus
stops somewhere between 400 yards away
from A and 600 yards away from A?

600           600
1         200

 f ( x)dx  400 2000 dx  2000  0.1
400
   What is the probability that the bus stops
within 200 yards away from point B?

2000            2000
1         200

1800
f ( x)dx  
1800
2000
dx 
2000
 0.1
   What is the probability that the bus stops half
way between A and B.

1000                1000
1        1000

0
f ( x)dx     
0
2000
dx 
2000
 0.5
Given that a continuous r.v. follows a uniform
distribution with pdf:

 1
       ,a  X  b
f ( x)   b  a
0, elsewhere

ab
E( X ) 
2
(b  a ) 2
Var ( X ) 
12
   Let T be the time when a STAT225 student
turned in his/her exam 1 hour after the exam
started. Suppose this time is uniformly/evenly
distributed between 9pm and 9:30pm.
   What is the pdf of T?
   What is the probability that a student turned
in the exam between 9:10pm and 9:25pm?
   What is the mean and standard deviation of
T?
   What is the probability that a student turned
in the exam at 9:30pm?

   What is the probability that a student turned
in the exam by 9:30pm?

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