1. ## probability density

Please give me some guidlance on the following problem :

Let
X be a random variable such that its values lie in the interval [1 1] and consider the function

f
: [1 1]- R f (x) = ((x\2) +1 ) * e^(x-1) .

a) Prove that
f can be the probability density of X.

b) Compute the expected value of
X.

Many thanks,

f
: [1 1]- R f (x) = ((x\2) +1 ) * e^(x-1) .

3. Let X be a random variable such that its values lie in the interval [1 1] and consider the function

f: [-1,1] in the set of real numbers and we have
f(x) = ((x\2) +1 ) * e^(x-1)

a) Prove that f can be the probability density of X.
b) Compute the expected value of
X.

Hope now it's clear for you Unfortunately it's not for me . Please give me some guidance.

Let X be a random variable such that its values lie in the interval [1 1] and consider the function

f: [-1,1] in the set of real numbers and we have
f(x) = ((x\2) +1 ) * e^(x-1)

a) Prove that f can be the probability density of X.
b) Compute the expected value of X.

Hope now it's clear for you Unfortunately it's not for me . Please give me some guidance.

a) You need to show that $\displaystyle \int_{-1}^{1} \left( \frac{x}{2} + 1\right) e^{x-1} \, dx = 1$.

b) You need to calculate $\displaystyle \int_{-1}^{1} x \left( \frac{x}{2} + 1\right) e^{x-1} \, dx$.

In both cases integration by parts is a way of doing this.