Question

Just the last part please :)

Consider the following distributions for a random variable X. In each case, match the distribution with the expectation of Y= 10-2x2 Poisson distribution with ?=5-50 Binomial(n= 10,p=0.3) Geometric(p-0.6) 12.2 4 0 x-oo! Choose.. ?

Answer 1

$E(X^2) = \int_{0}^{\infty }x^2 *f(x)dx = \int_{0}^{\infty }x^2 * \frac{1}{2}e^{-\frac{x}{2}}dx$

$\Rightarrow E(X^2) = \int_{0}^{\infty }x^2 * \frac{1}{2}e^{-\frac{x}{2}}dx$

$\Rightarrow E(X^2) = \frac{1}{2}\int_{0}^{\infty }x^2 *e^{-\frac{x}{2}}dx$

$\Rightarrow E(X^2) = \frac{1}{2} * \frac{\Gamma (3)}{(\frac{1}{2})^3}$

$\Rightarrow E(X^2) = \frac{1}{2} * \frac{2!}{(\frac{1}{2})^3}$

$\Rightarrow E(X^2) = \frac{2!}{(\frac{1}{2})^2}$

$\Rightarrow E(X^2) = 2*2^2 = 8$

Now,