For a random variable $X$ with mean $\mu$ and standard deviation $\sigma$, Chebyshev's inequality states that for $k>1$,

$P(|X-\mu|\geq k\sigma)\leq \frac{1}{k^2}$

$P(|X-\mu|\geq k\sigma)\leq \frac{1}{k}$

$P(|X-\mu|\geq k\sigma)\geq \frac{1}{k^2}$

$P(|X-\mu|\leq k\sigma)=\frac{1}{k^2}$

For any random variables $X$ and $Y$ with finite means, what is $E[X+Y]$?

$E[X]+E[Y]+2\operatorname{Cov}(X,Y)$

If $X$ and $Y$ are independent, which formula gives $\operatorname{Var}(X+Y)$?

$\operatorname{Var}(X)+\operatorname{Var}(Y)$

$\operatorname{Var}(X)\operatorname{Var}(Y)$

$\operatorname{Var}(X)-\operatorname{Var}(Y)$

$\operatorname{Var}(X)+\operatorname{Var}(Y)+2E[X]E[Y]$

What is the general formula for $\operatorname{Var}(X+Y)$?

$\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)$

$\operatorname{Var}(X)+\operatorname{Var}(Y)$

$\operatorname{Var}(X)\operatorname{Var}(Y)+\operatorname{Cov}(X,Y)$

$\operatorname{Var}(X)+\operatorname{Var}(Y)-\operatorname{Cov}(X,Y)$

ECAP790 Unit 4: Moments - MCQ Practice | MCA - ODL

1 $For a random variable with mean and standard deviation, Chebyshev's inequality states that for,$

Chebyshev Inequality Easy

A.

B.

C.

D.

2 $According to Chebyshev's inequality, at least what proportion of observations lies within standard deviations of the mean?$

Chebyshev Inequality Easy

A.

B.

C.

D.

3 $Which condition is required to apply Chebyshev's inequality to a random variable?$

Chebyshev Inequality Easy

A.

It must be a discrete random variable

B.

It must have a symmetric, continuous, and bell-shaped probability distribution

C.

It must be normally distributed

D.

It must have finite mean and variance

4 $Which expression defines a mixed raw moment of order for random variables and ?$

Moments of Two or More Random Variables Easy

A.

B.

C.

D.

5 $Which expression represents the covariance of and ?$

Moments of Two or More Random Variables Easy

A.

B.

C.

D.

6 $If and are independent and their moments exist, what is ?$

Moments of Two or More Random Variables Easy

A.

B.

C.

D.

7 $For any random variables and with finite means, what is ?$

Moments of Sums of Random Variables Easy

A.

B.

C.

D.

8 $If and are independent, which formula gives ?$

Moments of Sums of Random Variables Easy

A.

B.

C.

D.

9 $What is the general formula for ?$

Moments of Sums of Random Variables Easy

A.

B.

C.

D.

10 $How is the moment generating function of a random variable defined?$

Moment Generating Function Easy

A.

B.

C.

D.

11 $If the moment generating function exists near, how is the first raw moment obtained?$

Moment Generating Function Easy

A.

B.

C.

D.

12 $Which derivative of evaluated at gives ?$

Moment Generating Function Easy

A.

B.

C.

D.

13 $What is the value of for any random variable whose MGF exists?$

Properties of moment generating function Easy

A.

B.

C.

D.

14 $If and are independent, what is the MGF of ?$

Properties of moment generating function Easy

A.

B.

C.

D.

15 $If two random variables have the same MGF in an interval around, what can be concluded?$

Properties of moment generating function Easy

A.

They have equal means but necessarily have different variances and distribution shapes

B.

They have the same distribution

C.

They have the same observed sample

D.

They must both follow a normal distribution

16 $How is the cumulant generating function defined using the MGF?$

Cumulants Easy

A.

B.

C.

D.

17 $What does the first cumulant of a random variable represent?$

Cumulants Easy

A.

The fourth central moment adjusted using the variance

B.

The skewness

C.

The mean

D.

The variance

18 $What does the second cumulant of a random variable represent?$

Cumulants Easy

A.

The sum of the mean and the third raw moment

B.

The mean

C.

The standard deviation

D.

The variance

19 $What is the th raw moment of a random variable ?$

Raw and central moments Easy

A.

B.

C.

D.

20 $Which statement about the first and second central moments is correct?$

Raw and central moments Easy

A.

They are and, respectively

B.

They are and, respectively

C.

They are both equal to the standard deviation whenever the distribution is symmetric

D.

They are and, respectively

21 $A random variable has mean and variance . Using Chebyshev's inequality, what is the minimum guaranteed value of ?$

Chebyshev Inequality Medium

A.

B.

C.

D.

22 $A distribution has mean and variance . Which interval is guaranteed by Chebyshev's inequality to contain at least of the observations?$

Chebyshev Inequality Medium

A.

B.

C.

D.

23 $If has mean and variance, what upper bound does Chebyshev's inequality give for ?$

Chebyshev Inequality Medium

A.

B.

C.

D.

24 $Suppose,, and . What is ?$

Moments of Two or More Random Variables Medium

A.

B.

C.

D.

25 $Independent random variables and satisfy and . Find the mixed moment .$

Moments of Two or More Random Variables Medium

A.

B.

C.

D.

26 $Given,, and, find .$

Moments of Two or More Random Variables Medium

A.

B.

C.

D.

27 $Random variables and have standard deviations and, respectively, with correlation . Find .$

Moments of Sums of Random Variables Medium

A.

B.

C.

D.

28 $Independent random variables and have means and and variances and . What is ?$

Moments of Sums of Random Variables Medium

A.

B.

C.

D.

29 $Independent random variables and have third central moments and, respectively. What is the third central moment of ?$

Moments of Sums of Random Variables Medium

A.

B.

C.

D.

30 $The moment generating function of is . Which distribution does have?$

Moment Generating Function Medium

A.

B.

C.

D.

31 $A random variable has MGF . What is the second raw moment ?$

Moment Generating Function Medium

A.

B.

C.

D.

32 $Suppose . What is ?$

Moment Generating Function Medium

A.

B.

C.

D.

33 $If and has MGF, which expression is the MGF of ?$

Properties of moment generating function Medium

A.

B.

C.

D.

34 $If and are independent and have the same MGF, what is the MGF of ?$

Properties of moment generating function Medium

A.

B.

C.

D.

35 $Suppose the MGFs of and exist on an open interval containing and satisfy throughout that interval. What follows?$

Properties of moment generating function Medium

A.

and are equal for every outcome

B.

and have the same mean only

C.

and have the same distribution

D.

and must be independent

36 $A random variable has cumulant generating function . Which pair gives its first two cumulants?$

Cumulants Medium

A.

B.

C.

D.

37 $If, what is its fourth cumulant ?$

Cumulants Medium

A.

B.

C.

D.

38 $Independent variables and have third cumulants and . Find .$

Cumulants Medium

A.

B.

C.

D.

39 $A random variable has,, and . What is its third central moment?$

Raw and central moments Medium

A.

B.

C.

D.

40 $Suppose,,, and . Find the fourth central moment .$

Raw and central moments Medium

A.

B.

C.

D.

41 $A random variable has and . What is the best lower bound for obtained directly from Chebyshev's inequality?$

Chebyshev Inequality Hard

A.

B.

C.

D.

42 $Suppose and . What is the smallest for which Chebyshev's inequality guarantees ?$

Chebyshev Inequality Hard

A.

B.

C.

D.

43 $Let and define . Without additional distributional assumptions, what upper bound does Chebyshev's inequality give for ?$

Chebyshev Inequality Hard

A.

B.

C.

D.

44 $Random variables and satisfy,,,, and . Find .$

Moments of Two or More Random Variables Hard

A.

B.

C.

D.

45 $Let be jointly normal with zero means,,, and . What is ?$

Moments of Two or More Random Variables Hard

A.

B.

C.

D.

46 $Independent random variables and satisfy and . Evaluate .$

Moments of Two or More Random Variables Hard

A.

B.

C.

D.

47 $Suppose,, and . Find .$

Moments of Sums of Random Variables Hard

A.

B.

C.

D.

48 $Independent random variables and have third central moments and . What is the third central moment of ?$

Moments of Sums of Random Variables Hard

A.

B.

C.

D.

49 $Independent centered random variables and satisfy,,, and . Find .$

Moments of Sums of Random Variables Hard

A.

B.

C.

D.

50 $The first four raw moments of are,,, and . What is the fourth central moment ?$

Raw and central moments Hard

A.

B.

C.

D.

51 $A random variable has moment generating function for . Find .$

Moment Generating Function Hard

A.

B.

C.

D.

52 $The maximal interval on which is finite is . If, which expression and maximal interval describe ?$

Moment Generating Function Hard

A.

on

B.

on

C.

on

D.

on

53 $A random variable has moment generating function . What is ?$

Moment Generating Function Hard

A.

B.

C.

D.

54 $Let, where . Which statement correctly describes the moments and moment generating function of ?$

Properties of moment generating function Hard

A.

All raw moments exist, but for every .

B.

Only finitely many raw moments exist, while is finite near zero.

C.

No positive raw moments exist, although is finite for negative .

D.

All raw moments exist, and is finite on an interval around zero.

55 $Suppose for every in some open interval containing, and both MGFs are finite there. What follows?$

Properties of moment generating function Hard

A.

and are equal for every outcome.

B.

and must be mutually independent.

C.

and have the same distribution.

D.

and have only equal first moments.

56 $For random variables whose MGFs are finite near zero, which statement about is generally valid?$

Properties of moment generating function Hard

A.

Independence is necessary, but it is not sufficient.

B.

Independence is sufficient, but the converse need not hold.

C.

The equality holds exactly when covariance is zero.

D.

The equality is equivalent to pairwise independence.

57 $Assume and define a tilted distribution by . What is the MGF of under ?$

Properties of moment generating function Hard

A.

B.

C.

D.

58 $Let, and let denote the th cumulant of . Which transformation rule is correct?$

Cumulants Hard

A.

and for

B.

and for

C.

and for

D.

and for

59 $Let and be independent, and define . What are ?$

Cumulants Hard

A.

B.

C.

D.

60 $A random variable has mean, variance, and third central moment . What is its third raw moment ?$

Raw and central moments Hard

A.

B.

C.

D.

Unit 4: Moments - Practice Quiz

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