Question

Assume a Poisson random variable has a mean of 5 successes over a 125-minute period.

Answer 1

Answer:

a

$\mu = 25$

b

$\lambda =0.04$

c

$P(X > 65 ) =0.0742$  

Step-by-step explanation:

From the question we are told that

  The  number of success is  n =  5

   The time duration is  N  =  125 minutes

   

Generally the mean of the random variable is mathematically represented as

       $\mu = \frac{N}{n}$

=>    $\mu = \frac{125}{5}$

=>    $\mu = 25$

Generally the rate parameter is mathematically represented as

      $\lambda = \frac{1}{\mu}$

=>   $\lambda = \frac{1}{25}$

=>   $\lambda =0.04$

Generally the cumulative distribution function for exponential distribution function is  

   $F(x) = \left \{0 \ \ \ \ \ \ x \le 0}} \atop {1 -e^{-\lambda t }\ x>0}} \right.$

Generally the probability that the time to success will be more than 63 minutes is mathematically represented as

 $P(X > 65 ) = 1 - P(X \le 65)$

Here

    $P(X \le 65) = 1 -e^{-65 * \lambda }$  

=>       $P(X \le 65) = 1 -e^{-65 * 0.04 }$    

=>       $P(X \le 65) = 0.92573$    

So

     $P(X > 65 ) = 1 - 0.92573$  

     $P(X > 65 ) =0.0742$