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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}
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