<PRE><B>Question 1206029</B>
.
A plumber tells you she will arrive between 8 am and noon. Assume any time in this interval 
is an equally likely arrival time. Let X= arrival time (minutes after 8 am). 
State the distribution that X follows in proper mathematical notation.


(a) Find the probability density function f(x) and sketch a graph of it in your notes.


(b) What is the probability she arrives before 9 am? 
Pose this question with mathematical notation and compute the answer.


(c) What is the probability she arrives between 10:15 and 11 am? 
Pose this question with mathematical notation and compute the answer.


(d) It is now 9:30. What is the probability that she arrives in the next 30 minutes? 
Pose this question with mathematical notation and compute the answer.
~~~~~~~~~~~~~~~~~~~~~



&lt;pre&gt;
This probability distribution as described in the problem, is called a &lt;U&gt;UNIFORM&lt;/U&gt; distribution.


(a)  Find the probability density function f(x) and sketch a graph of it in your notes.


     The formula for the density distribution is  f( x ) = {{{1/240}}} = constant.

     Here x represents any ONE minute time interval after 8 am; 
     240 represents 4 hours from 8 am to noon in minutes, 240 = 60 minutes * 4 hours.



(b)  What is the probability she arrives before 9 am? 
     Pose this question with mathematical notation and compute the answer.


     P(x &lt;= 9 am) = {{{60/240}}} = {{{1/4}}}, 

     saying that the probability to arrive between 8 am and 9 am is 1/4. 



(c)  What is the probability she arrives between 10:15 and 11 am? 
     Pose this question with mathematical notation and compute the answer.


     P(10:15 am x &lt;= 11 am) = {{{45/240}}} = {{{3/16}}}, 

     saying that the probability to arrive between 10:15 am and 11:00 am is 3/16. 



(d) It is now 9:30. What is the probability that she arrives in the next 30 minutes? 
    Pose this question with mathematical notation and compute the answer.


     P(x &lt;= 30 minutes given that it is 9:30 now) = {{{30/150}}} = {{{1/5}}}, 

     saying that the probability to arrive next 30 minutes given that it is 9:30 now
     is 1/5.   

     It is BECAUSE there are only 2 hours 30 minutes, or 150 minutes, from now till noon.
&lt;/pre&gt;

At this point, the problem is solved in full - I answered all questions,
having provided you all necessary explanations.



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