Moments and Moment Generating Function

The expectation of a random variable X (continuous or discrete) is given by

• $\sum_{ }^{ }xf(x),\ \int\ xf(x)$

• $∑x^2f(x),\ ∫x^2f(x)$

• $∑f(x),\ ∫f(x)$

• $∑xf(x^2),\ ∫xf(x^2)$

Mean of a random variable X is given by

• $E(X)$

• $E(X^2)$

• $E(X^2)–(E(X))^2$

• $(E(X))^2$

Variance of a random variable X is given by

• $E(X)$

• $E(X^2)$

• $E(X^2)–(E(X))^2$

• $(E(X))^2$

Let f(x) be the pdf of the random variable X, then mean of a constant ‘a’ is

• 0

• a

• $\frac{a}{2}$

• 1

Variance of a constant ‘a’ is

• 0

• a

• 1

• Both a and b

Find the expectation of a random variable X if $f(x)=ke^{-x}$  for  $x\ >\ 0$  and 0 otherwise.

• 0

• 1

• 2

• 3

Find the mean of a random variable X if $f(x)=x–\ \frac{5}{2}$  for  $0<x<1$  and  $2x$  for  $1<x<2$  and 0 otherwise.

• 2.5

• 2.75

• 3.5

• 3.75

What is moment generating function?

• $M_x(t)=E(e^{tx})$

• $M_x(t)=E(e^{-tx})$

• $M_x(t)=E(e^{2tx})$

• $M_x(t)=E(e^{-x})$

Find the Moment Generating Function of $f\left(x\right)\ =\ x$ for  $0<\ x<1$  and  $2-x$  for  $1<x<2$  and 0 otherwise.

•  $\left(\frac{e^t-1}{t}\right)^2$  

•  $\left(\frac{e^{-t}-1}{t}\right)^2$  

•  $\left(\frac{e^{2t}-1}{t}\right)^2$  

•  $\left(\frac{e^t-1}{t}\right)^{ }$  

Find the mean of a continuous random variable X if $f(x)=2e^{-x}$  for  $x>0$  and  $-e^x$  for  $x<0$  .

• 0

• 1

• 2

• 3