Let  ƒ : R → R an applicationdefined for all x, by:  

Where λ € R and µ > 0   

1- Determine kas a function of µ such that ƒ is the probability density function of a random variable X.

2- What is the distribution function of X?

3- If we define Y=X-λ/µ. What is the distribution of Y?

4- Calculate E(X) and V(X)

5- Evaluate P (| Y | ≥ 2). Compare with P (| U | ≥ 2) where U successively follows a distribution  N (0, 1), a distribution of Student at 20 degrees of freedom, a Cauchy distribution of density ƒ(u)=1/∏(1+x²), a double exponential distribution of density ƒ(x) = exp {-x-e^-x}