Consider a random variable X with probability density function
\(f(x) = \begin{cases} ke^{|x|},for & -\infty<x<\infty \\ 0,& otherwise \\ \end{cases}\).
Find the value of k,mean and variance.

A) \$2/3,E(X)=2,Var(X) =7\$
B) \$1/3,E(X)=1,Var(X) = 2\$
C) \$1/2,E(X)=0,Var(X) = 2\$
D) \$2/5,E(X)=2,Var(X) = 4\$

Find the mean and variance of the distribution
\(f(x) = \begin{cases} 3e^{-3x}, 0<x<\infty \\ 0,& otherwise \\ \end{cases}\)

A) \$E(X)=1/4,V(X)=1/8\$
B) \$E(X)=1/3,V(X)=1/9\$
C) \$E(X)=1/4,V(X)=1/9\$
D) \$E(X)=1/3,V(X)=1/8\$

An insurance company insures a large number of homes. The insured value, X, of a randomly selected home is assumed to follow a distribution with density function.
\(f(x) = \begin{cases} 3/x^4,for & x > 1 \\ 0,& otherwise \\ \end{cases}\)
Given that a randomly selected home is insured for at least 1.5, calculate the probability that it is insured for less than 2.

A) \$21/47\$
B) \$28/33\$
C) \$35/67\$
D) stem:[37/64

The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable X with probability density function.
\(f(x) = \begin{cases} k(9-x),for & 0≤x≤9 \\ 0,& otherwise \\ \end{cases}\)
Find the mean lifetime of a component.

A) 8
B) 5
C) 2
D) 3

Consider a random variable X with probability density function
\(f(x) = \begin{cases} k(4x-x^3),for & 0≤x≤2 \\ 0,& otherwise \\ \end{cases}\)
.Find k,mean.

A) \$k=1/14,19/14\$
B) \$k=1/8,16/13\$
C) \$k=1/4,16/15\$
D) \$k=1/2,12/13\$