Consider a andom variable X with probability density function
\(f(x) = \begin{cases} 4x^3,for & 0≤x≤1 \\ 0,& otherwise \\ \end{cases}\).
Find mean and variance.

A) \$E(X)=2/5,Var(X) = 4/75\$
B) \$E(X)=2/5,Var(X) = 2/45\$
C) \$E(X)=4/6,Var(X) = 2/45\$
D) \$E(X)=4/5,Var(X) = 2/75\$

A continuous random variable X has a probability density function
\(f(x)= \begin{cases} e^-x,for & 0<x<\infty\\ 0,& otherwise \\ \end{cases}\).
Then P(X > 1)

A) 0.354
B) 0.352
C) 0.368
D) 0.313

Given P.D.F. of a continuous random variable x as
\(f(x) = \begin{cases} x^2/3,for & -1≤x≤2 \\ 0,& otherwise \\ \end{cases}\).
Determine the C.D.F. of x and find P(1<x<2)

A) f(x)=\$(x^8+1)/9\$,P(1<x<2)=\$13/9\$
B) f(x)=\$(x^3+1)/9\$,P(1<x<2)=\$7/9\$
C) f(x)=\$(x^3+1)/7\$,P(1<x<2)=\$4/7\$
D) f(x)=\$(x^5+1)/9\$,P(1<x<2)=\$5/9\$

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

A) 2,\$1/2\$
B) 5,\$1/2\$
C) 2,\$1/8\$
D) 5,\$1/8\$

A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is
f(x) = \$1/25\$,for 0<x<25
Obtain and interpret the expected value of the random variable X.

A) 11.3
B) 14.1
C) 12.5*
D) 15.4