Free Printable Moments and Moment Generating Function

1.

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

a)

$\sum_{ }^{ }xf(x),\ \int\ xf(x)$

b)

$∑x^2f(x),\ ∫x^2f(x)$

c)

$∑f(x),\ ∫f(x)$

d)

$∑xf(x^2),\ ∫xf(x^2)$

2.

Mean of a random variable X is given by

a)

$E(X)$

b)

$E(X^2)$

c)

$E(X^2)–(E(X))^2$

d)

$(E(X))^2$

3.

Variance of a random variable X is given by

a)

$E(X)$

b)

$E(X^2)$

c)

$E(X^2)–(E(X))^2$

d)

$(E(X))^2$

4.

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

a)

0

b)

a

c)

$\frac{a}{2}$

d)

1

5.

Variance of a constant ‘a’ is

a)

0

b)

a

c)

1

d)

Both a and b

6.

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

a)

0

b)

1

c)

2

d)

3

7.

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.

a)

2.5

b)

2.75

c)

3.5

d)

3.75

8.

What is moment generating function?

a)

$M_x(t)=E(e^{tx})$

b)

$M_x(t)=E(e^{-tx})$

c)

$M_x(t)=E(e^{2tx})$

d)

$M_x(t)=E(e^{-x})$

9.

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.

a)

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

b)

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

c)

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

d)

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

10.

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

a)

0

b)

1

c)

2

d)

3

11.

The expectation of a random variable X, (E(X)) can be written as

a)

$\frac{d}{dt}[M_X(t)](t=0)\$

b)

$\frac{d}{dx}[M_X(t)](t=0)$

c)

$\frac{d^2}{dt^2}[M_X(t)](t=0)$

d)

$\frac{d^2}{dx^2}[M_X(t)](t=0)$

Moments and Moment Generating Function

Total questions: 11

Worksheet time: 54mins

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