Question

The number of cracks that need repair in a section of the interstate highway follows a Poisson distribution with a mean of two cracks per mile.
1. Determine the probability mass function of the number of cracks (X) in 5 miles of highway.
2. Find the probability that there are at least five cracks in 5 miles of highway that require repair.
3. Calculate the mean and variance of X.

Answer 1

Answer:

KENDRIYA VIDYALAYA NO.1 UPPAL HYDERABAD

Answer 2

Given a poisson distribution with mean of 2 cracks/mile in a 5 mile highway

Explanation:

1.The probability mass function of a poisson distribution is, $P(x)=\frac{\lambda^xe^{-\lambda}}{x!}$  

  for a random variable 'x' and λ being the mean which is $2$ cracks per mile

2.Here these $5$ miles are considered with $2$ cracks per mile having total of      

  $(5x2=10) 10$ cracks possible over the said $5$ miles.

3. hence, random variable x (being no. of cracks possible) ranges till $10$.

4. probability that there are at least $5$ cracks means $x\geq 5$,

         $P(x\geq 5)= P(x=5)+ P(x=6)+P(x=7)+ P(x=8)+P(x=9)+P(x=10)$

  equating these values for $x$ in $P(x)$ from first point we get,

       $P(x\geq 5)=0.036 + 0.012+ 0.003+0.0008+0.00019+ 0.00076$                                      

       $P(x\geq 5)=0.05275$  

5. as already given for the above poisson distribution,

     mean = $2$

6. for a poisson distribution mean and variance have equal value ,

    variance=  $\lambda=2$