Transcript Exponential Distribution

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Recall your experience when you take an
elevator.
Think about usually how long it takes for the
elevator to arrive.
Most likely, the experience may be it
frequently comes in a short while and once in
a while, it may come pretty late.
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In another word, if we want to use a random
variable to measure the waiting time for
elevator to come, we can say that:
◦ 1. It must be continuous.
◦ 2. Smaller values have larger probability and larger
values have smaller probability.
◦ Think about Geometric distribution, is there any
similarities?
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Usually, exponential distribution is used to
describe the time or distance until some
event happens.
It is in the form of:
f ( x) 
1
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
e
x
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◦ where x ≥ 0 and μ>0. μ is the mean or expected
value.
f ( x)   e
In this case,
 
 x
1
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Then the mean or expected value is
1
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1
0.8
0.6
R(x)
0.4
0.2
0
0
x
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We also use CDF to find probabilities under
exponential distribution.
x0
P( x  x0 )   f ( x)dx  1  e
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
x0

0
Or
xb
P( xa  x  xb )   f ( x)dx  P( x  xb )  P( x  xa )
xa
xb
xb
xa
xa
0
0
  f ( x)dx   f ( x)dx   f ( x)dx  1  e

xb
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 (1  e

xa

)e

xa

e

xb
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On average, it takes about 5 minutes to get
an elevator at Math building. Let X be the
waiting time until the elevator arrives. (Let’s
use the form with μ here)
◦ Find the pdf of X.
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2. What is the probability that you will wait
less than 3 minutes?
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3. What is the probability that you will wait
for more than 10 minutes?
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What is the probability that you will wait for
more than 7 minutes?
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Given that you already wait for more than 3
minutes, what is the probability that you will
wait for more than 10 minutes?
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That is a very interesting and useful property
for exponential distribution.
It is called “Memorylessness” or simply “Lack
of memory”.
In mathematical form: P( X  s  t | X  s)  P( X  t )
Therefore, P(wait more than 10 minutes| wait
more than 3 minutes)=P(wait more than 7+3
minutes| wait more than 3 minutes)=P(wait
more than 7 minutes)
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E(X)= μ or 1
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1
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Var(X)= μ2 or  2
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Poisson is a discrete random variable that
measures the number of occurrence of some
given event over a specific interval (time,
distance)
Exponential describes the length of the
interval between occurrence.
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Example II:
A storekeeper estimated that on average,
there are 10 customers visiting his store
between 10am and 12pm everyday. However,
it has been more than 30 minutes since the
last customer visited. What is the probability
for that?
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If we know that there are on average 10
customers visiting a store within 2-hour
interval, then the average time between
customers’ arrival is: 120/10=12 minutes.
Therefore, the time interval between
customer visits follows an exponential
distribution with mean=12 minutes.
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Given that the storekeeper has not got any
customers for more than 30 minutes, what is
the probability that the storekeeper will still
have no customer for another 15 minutes or
more?
1.
2.
3.
Given a random variable X, a percentile
means a specific value of X, say x0.
Usually, when we say p-th percentile, we
mean there is a value x0 such that p% of the
values of X fall below x0.
A special case is the median, which is 50-th
percentile. That means 50% of the values of
X fall below it.
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We have learned that given some data points,
how to find the percentiles. In those cases,
the number of data points is finite or limited.
Now we turn to a different question, that is,
to look for a percentile for a continuous
random variable X, with NO data points given
but there are infinitely many possible values.
For example, if X~Uniform(0, 20), what is the
median, 25%, 75% percentile?
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X~Uniform(0, 20), what is the median, 25%,
75% percentile?
We can work out this kind of problem in the
form of solving an equation.
Let x0 be the median, then P(X<x0)=0.5.
(hint: 1. think about the shape of the pdf of a
uniform random variable; 2. how do we find
probability for a continuous random
variable?)
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Also, how do we find the 25th and 75th
percentile of X?
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What is the mean of X?
Compare the mean and median of X, what
can we find?
Can we tell that from the shape of the pdf of
X?
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Another example:
Y~Exp(5), find the median, 25th and 75th
percentile.
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Compare the mean and median of Y, are they
the same?
Why?