Answer:
a) 
And let X our random variable who represent the "occurrence of structural loads over time" we know that:
And the expected value is 
So we expect 4 number of loads in the 2 year period.
b) ![P(X(2) >6) = 1-P(X(2)\leq 6)= 1-[P(X(2) =0)+P(X(2) =1)+P(X(2) =2)+...+P(X(2) =6)]](https://tex.z-dn.net/?f=%20P%28X%282%29%20%3E6%29%20%3D%201-P%28X%282%29%5Cleq%206%29%3D%201-%5BP%28X%282%29%20%3D0%29%2BP%28X%282%29%20%3D1%29%2BP%28X%282%29%20%3D2%29%2B...%2BP%28X%282%29%20%3D6%29%5D)
![P(X(2) >6) = 1- [e^{-4}+ \frac{e^{-4}4^1}{1!}+ \frac{e^{-4}4^2}{2!} +\frac{e^{-4}4^3}{3!} +\frac{e^{-4}4^4}{4!}+\frac{e^{-4}4^5}{5!}+\frac{e^{-4}4^6}{6!}]](https://tex.z-dn.net/?f=%20P%28X%282%29%20%3E6%29%20%3D%201-%20%5Be%5E%7B-4%7D%2B%20%5Cfrac%7Be%5E%7B-4%7D4%5E1%7D%7B1%21%7D%2B%20%5Cfrac%7Be%5E%7B-4%7D4%5E2%7D%7B2%21%7D%20%2B%5Cfrac%7Be%5E%7B-4%7D4%5E3%7D%7B3%21%7D%20%2B%5Cfrac%7Be%5E%7B-4%7D4%5E4%7D%7B4%21%7D%2B%5Cfrac%7Be%5E%7B-4%7D4%5E5%7D%7B5%21%7D%2B%5Cfrac%7Be%5E%7B-4%7D4%5E6%7D%7B6%21%7D%5D)
And we got: 
c) 
We can apply natural log in both sides and we got:
If we multiply by -1 both sides of the inequality we have:
And if we divide both sides by 2 we got:
And then we can conclude that the time period with any load would be 0.8047 years.
Step-by-step explanation:
Previous concepts
The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:
The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution"
Solution to the problem
Let X our random variable who represent the "occurrence of structural loads over time"
For this case we have the value for the mean given 
and we can solve for the parameter 
like this:
So then 
X follows a Poisson process
Part a
For this case since we are interested in the number of loads in a 2 year period the new rate would be given by:
And let X our random variable who represent the "occurrence of structural loads over time" we know that:
And the expected value is
So we expect 4 number of loads in the 2 year period.
Part b
For this case we want the following probability:
And we can use the complement rule like this
And we can solve this like this using the masss function:
And we got:
Part c
For this case we know that the arrival time follows an exponential distribution and let T the random variable:
The probability of no arrival during a period of duration t is given by:
And we want to find a value of t who satisfy this:
We can apply natural log in both sides and we got:
If we multiply by -1 both sides of the inequality we have:
And if we divide both sides by 2 we got:
And then we can conclude that the time period with any load would be 0.8047 years.
