Download Find the cumulative distribution function (cdf) for an exponential

Test 2 – November 4th
Poisson Distribution
Continuous Random Variables –
Probability Density Functions (pdf)

f  0,

f ( x)dx  1;

b
P (a  X  b)   f ( x)dx
a
Cumulative Distribution Functions (cdf) - properties
The cdf of a continuous random variable is continuous.
Mean, variance, & standard deviation of continuous random variables
Conditional probabilities
Specific continuous distributions:
Uniform
Exponential (memoryless)
Normal
Gamma – the amount of time until r occurrences of a Poisson r.v.
Beta
Y = aX + b
Find the mean & variance of Y given the mean & variance of X.
___________________________________________________________________________________________________
Gamma Distribution
In a Poisson Process, the length of time before the event occurs t times has a Gamma distribution with parameters
( t, where  is the mean of the Poisson distribution.
Example: Suppose that, on average, the number of -particles emitted from a radioactive substance is four every second.
What is the probability that it takes at least two seconds before the next three -particles are emitted?
Let N(t*) denote the number of -particles emitted from a radioactive substance in the time interval [0, t*], t*>0. It is
reasonable that N(t*) is a Poisson process with when t* = 1. X, the time between now and the third -particle is
emitted, has a gamma distribution (3, 4). We want P( X  2) .
Exercises.
1. Show that (t)
= (t-1)(t-1).
2. Customers arrive at a restaurant at a Poisson rate of 12 per hour.
a) What is the probability that at least two hours elapse before the 20th customer arrives?
b) If the restaurant makes a profit only after 30 customers have arrived, what is the expected length of time until the
restaurant starts to make a profit?
3.
The response times on an online computer terminal have approximately a gamma distribution with mean 4 seconds and
variance 8 seconds squared. Write the probability density function for the response time. What is the probability that
the response time would exceed 5 seconds?
4. What is the relationship between the Gamma distribution and the exponential distribution?
Math 309
Old Test 3
Show all work in order to receive credit.
0 x2
elsewhere.
1. Let f ( x)  kx

0,
a)
b)
c)
d)
Carter
Name________________________
.
(20 pts)
Find k so that f(x) is a valid probability density function for a random variable X.
Find P( X < 0.3)
Find P( X < 0.3 | X < 1)
Find the mean and variance of X.
2. Emily’s commute to school varies randomly between 22 and 29 minutes. If she leaves at 7:35 a.m. for an 8 a.m. class,
what is the probability that she is on time?
(7 pts)
3. Use the probability density function to find the cumulative distribution function (cdf) for an exponential random variable
with mean  .
(7 pts)
4. The number of accidents in a factory can be modeled by a Poisson process averaging 2 accidents per week.
a) Find the probability that the time between successive accidents is more than 1 week.
b) Let W denote the waiting time until three accidents occurs. Find the mean, variance, and the probability density
function of W.
c) Find the probability that it is less than one week until the occurrence of three accidents. (Setting up the integral is
sufficient.)
(18 pts)
5. The annual rainfall in a certain region is normally distributed with mean 29.5 inches and standard deviation 2.5 inches.
a) Find the probability that the annual rainfall is between 28 and 29 inches.
b) How many inches of rain is exceeded only 1% of the time?
(14 pts)
6. The proportion of pure iron in certain ore samples has a beta distribution, with   3 and   1 .
a) Find the probability that one of these samples will have more than 40% pure iron.
b) Find the probability that exactly three out of four samples will have more than 40% pure iron.
7. Select one of the following to prove:
a) If X has an exponential distribution with
b) Let
  0.
   , then P(X > a+b | X > a) = P ( X > b).

If
Γ(α)   x α 1e  x dx , then (a+1) = a(a).
0
c)
Derive the mean of the Beta distribution

d) Show that

f ( x)dx  1 where f(x) is the pdf of the gamma distribution.

Answers:
x
1. k = 0.5; .0225; .09; 4/3, 2/9;
2. 3/7 3. F ( x) 
1
 e
t /
dt  1  e  x / if x  0, 0 if x  0;
0
4. Let T = time from occurrence of 1 accident until the next accident. T is exponential w/ lambda = 2


1

f ( w)  
w2 e 2 w
3
 (3)(.5 )
 0 w  0

P(T  1)   2e2t dt  e2 ; W is gamma w/ r = 3 and lambda = 2,
1
  3 / 2;  2 
w0
3
4
1
c)  4w2 e 2 w dw
0
5a) P(28 < X < 29) = P( -.6 < Z <-.2) = P(.2 < Z <.6) = .2257 - .0793 = .1464; 2.33 = (x – 29.5)/2.5 => x = 35.325
1
6a)
(4)
 (3)(1) x dx  .936 ; b)
0.4
2
C (4,3)(.936)3 (.064)1  .2099
(14 pts)
(10 pts)
1. Service calls come to a maintenance center according to a Poisson process and on the average 2.7calls come per minute.
a) Find the probability that no more than 4 calls come in any one minute.
b) Find the probability that more than 10 calls come in a 5-minute period.
(10)
2. Find the value of the constant, c, so that f(y) is a density function.
(10)
c y 0  y  4
f ( y)  
0
elsewhere
3a) Find F(x), given f(x).
b) Find P( 1 < X < 2 ).
 3x 2

f ( x)   16
0

2 x  2
(10)
elsewhere
7. A manufacturer has contracted to supply ball bearings. Product analysis reveals that the diameters are normally
distributed with a mean of 25.1 mm and a standard deviation of 0.2 mm.
(10)
a) What is the probability that a diameter exceeds 25.4 mm?
b) What is the minimum acceptable diameter if the smallest 13% of the diameters are unacceptable?
8.
The proportion of time that an industrial robot is in operation during a 40-hour week is a random variable with
probability density function
2 y 0  y  1
.
f ( y)  
0 elsewhere
a) Find the expected value and variance of Y.
b) If the profit for a week is given by X = 200Y – 60, find E(X) and V(X).
9a) Is the distribution function at the right for a discrete or
continuous random variable?
b) Find the density function or probability distribution of X.
Answers:
1) 0.8629, 0.7888
2) 3/16
3
3) 0 if X < -2; x /16 + ½ if –2  X  2; 1 if X > 2 ; 7/16
7) 0.0668, 24.874
8) 2/3, 1/18; 73.333, 2222.222
9) discrete; p(1)=0.2, p(2)=0.5, p(4)=0.3
(15)