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Unit 5: Relation Between Moments - Practice Quiz

ECAP790 60 Questions
0 Correct 0 Wrong 60 Left
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1 The first raw moment of a distribution is equal to its:

Relation between raw moments and central moments Easy
A. Variance
B. Mean
C. Skewness
D. Kurtosis

2 What is the value of the first central moment ?

Relation between raw moments and central moments Easy
A.
B.
C.
D.

3 Which formula relates the second central moment to raw moments?

Relation between raw moments and central moments Easy
A.
B. , which squares the entire difference
C.
D.

4 Central moments are moments measured about the:

Relation between raw moments and central moments Easy
A. Mean
B. Origin
C. Median
D. Mode

5 A change of origin has what effect on central moments?

Effect of change of origin and scale on moments Easy
A. They change sign
B. They become zero
C. No effect
D. Every central moment increases by the amount used to shift the origin

6 If , the variance of is:

Effect of change of origin and scale on moments Easy
A.
B.
C.
D.

7 If , how is the second central moment transformed?

Effect of change of origin and scale on moments Easy
A.
B.
C.
D.

8 For the transformation , the th central moment of is:

Effect of change of origin and scale on moments Easy
A.
B. , including the origin correction
C.
D.

9 Pearson's moment coefficient is defined as:

Pearsonian coefficients Easy
A.
B.
C.
D.

10 Pearson's coefficient is given by:

Pearsonian coefficients Easy
A.
B.
C.
D.

11 The moment coefficient of skewness is:

Pearsonian coefficients Easy
A. , using the fourth central moment
B.
C.
D.

12 What is the relationship between and ?

Pearsonian coefficients Easy
A.
B.
C.
D.

13 A perfectly symmetric distribution has a skewness of:

Measures of skewness Easy
A.
B.
C.
D.

14 A positively skewed distribution generally has a longer tail on the:

Measures of skewness Easy
A. Right side
B. Central part of the distribution around its arithmetic mean
C. Both sides
D. Left side

15 Karl Pearson's first coefficient of skewness is:

Measures of skewness Easy
A.
B.
C.
D.

16 Karl Pearson's second coefficient of skewness is:

Measures of skewness Easy
A.
B.
C.
D.

17 Kurtosis primarily describes the:

Kurtosis Easy
A. Direction of skewness
B. Peakedness and tail weight
C. Location of the mean
D. Exact difference between the largest and smallest observations

18 For a normal distribution, Pearson's kurtosis coefficient equals:

Kurtosis Easy
A.
B.
C.
D.

19 A distribution with is called:

Kurtosis Easy
A. Mesokurtic
B. Platykurtic
C. Leptokurtic
D. Symmetric with no measurable fourth central moment

20 The excess kurtosis is calculated as:

Kurtosis Easy
A.
B.
C.
D.

21 The first two raw moments of a distribution about the origin are and . What is the second central moment ?

Relation between raw moments and central moments Medium
A.
B.
C.
D.

22 For a distribution, , , and . Find the third central moment.

Relation between raw moments and central moments Medium
A.
B.
C.
D.

23 A distribution has mean , variance , and third central moment . What is its third raw moment ?

Relation between raw moments and central moments Medium
A.
B.
C.
D.

24 The first four raw moments of a distribution are , , , and . Find the fourth central moment.

Relation between raw moments and central moments Medium
A.
B.
C.
D.

25 A variable has mean and variance . If , what are the mean and variance of ?

Effect of change of origin and scale on moments Medium
A. Mean , variance
B. Mean , variance
C. Mean , variance
D. Mean , variance

26 The third central moment of is . If , what is the third central moment of ?

Effect of change of origin and scale on moments Medium
A.
B.
C.
D.

27 For , the first two raw moments are and . If , find the second raw moment of .

Effect of change of origin and scale on moments Medium
A.
B.
C.
D.

28 A variable has moment coefficient of skewness and coefficient of kurtosis . If , what are the corresponding coefficients for ?

Effect of change of origin and scale on moments Medium
A. ,
B. ,
C. ,
D. ,

29 For a distribution, , , and . Find Pearson's moment coefficients and .

Pearsonian coefficients Medium
A. ,
B. ,
C. ,
D. ,

30 A distribution has and a positive third central moment. What is its signed moment coefficient of skewness ?

Pearsonian coefficients Medium
A.
B.
C.
D.

31 If Pearson's coefficient of kurtosis is , what are the excess kurtosis and the corresponding classification?

Pearsonian coefficients Medium
A. , leptokurtic
B. , platykurtic
C. , platykurtic
D. , leptokurtic

32 A positively skewed distribution has variance and Pearson's coefficient . What is its third central moment?

Pearsonian coefficients Medium
A.
B.
C.
D.

33 A distribution has mean , mode , and standard deviation . Find Pearson's first coefficient of skewness.

Measures of skewness Medium
A.
B.
C.
D.

34 The mean, median, and standard deviation of a distribution are , , and , respectively. Find Pearson's second coefficient of skewness.

Measures of skewness Medium
A.
B.
C.
D.

35 For a distribution, , the median is , and . What is Bowley's coefficient of skewness?

Measures of skewness Medium
A.
B.
C.
D.

36 A distribution has open-ended extreme classes, making its mean and standard deviation unreliable. Which measure of skewness is most appropriate?

Measures of skewness Medium
A. The third-moment coefficient
B. Bowley's quartile coefficient
C. Pearson's mean-mode coefficient
D. Pearson's mean-median coefficient

37 A distribution has second central moment and fourth central moment . Find and classify the distribution.

Kurtosis Medium
A. , mesokurtic
B. , leptokurtic
C. , platykurtic
D. , leptokurtic

38 Which pair correctly gives the coefficient of kurtosis and excess kurtosis of a normal distribution?

Kurtosis Medium
A. ,
B. ,
C. ,
D. ,

39 A distribution has variance and coefficient of kurtosis . What is its fourth central moment?

Kurtosis Medium
A.
B.
C.
D.

40 A variable has excess kurtosis . If , what is the excess kurtosis and classification of ?

Kurtosis Medium
A. , leptokurtic
B. , platykurtic
C. , leptokurtic
D. , mesokurtic

41 The first four raw moments of about zero are , , , and . What is the fourth central moment ?

Relation between raw moments and central moments Hard
A.
B.
C.
D.

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

Relation between raw moments and central moments Hard
A.
B.
C.
D.

43 The first four moments of about an arbitrary origin are , , , and . Determine the fourth moment about the mean.

Relation between raw moments and central moments Hard
A.
B.
C.
D.

44 A distribution has mean and central moments , , and . Find its fourth raw moment.

Relation between raw moments and central moments Hard
A.
B.
C.
D.

45 Let , where and has nonzero variance. Which statement about moment-based shape coefficients is correct?

Effect of change of origin and scale on moments Hard
A. and
B. and
C. and
D. and

46 For , suppose and . Which pair gives and ?

Effect of change of origin and scale on moments Hard
A.
B.
C.
D.

47 The first four raw moments of are , , , and . If , what is the fourth raw moment of ?

Effect of change of origin and scale on moments Hard
A.
B.
C.
D.

48 For the coded variable , the first three raw moments are , , and . What are ?

Effect of change of origin and scale on moments Hard
A.
B.
C.
D.

49 A distribution has Pearson's coefficient and a negative third central moment. What is its signed moment coefficient of skewness ?

Pearsonian coefficients Hard
A.
B.
C.
D.

50 A nondegenerate distribution has Pearsonian coefficient of skewness . What is the smallest theoretically possible value of ?

Pearsonian coefficients Hard
A.
B.
C.
D.

51 For a nondegenerate random variable, when does equality hold in Pearson's inequality ?

Pearsonian coefficients Hard
A. Exactly when the distribution is a symmetric three-point distribution
B. Exactly when the distribution has zero moment skewness
C. Exactly when the distribution is a nondegenerate two-point distribution
D. Exactly when the distribution belongs to the normal family

52 A distribution has central moments , , and . Which tuple is correct?

Pearsonian coefficients Hard
A.
B.
C.
D.

53 A distribution has , , and Bowley's coefficient of skewness equal to . What is the median ?

Measures of skewness Hard
A.
B.
C.
D.

54 A variable has mean , median , and standard deviation . If , what is Pearson's second coefficient of skewness for ?

Measures of skewness Hard
A.
B.
C.
D.

55 For a distribution, , , and . What is Kelly's coefficient of skewness?

Measures of skewness Hard
A.
B.
C.
D.

56 A distribution has mean , standard deviation , and Pearson's first coefficient of skewness . Assuming the empirical relation holds exactly, which pair gives the median and Pearson's second coefficient?

Measures of skewness Hard
A.
B.
C.
D.

57 Independent identically distributed variables have finite variance and kurtosis . If their standardized sum is , what is ?

Kurtosis Hard
A.
B.
C.
D.

58 Independent variables and have variances and , and excess kurtoses and , respectively. What is the excess kurtosis of ?

Kurtosis Hard
A.
B.
C.
D.

59 Independent variables and have variances and , with kurtoses and . What is the kurtosis of ?

Kurtosis Hard
A.
B.
C.
D.

60 For a degenerate distribution concentrated at a single value, all central moments of positive order are zero. What is the mathematically appropriate conclusion about and ?

Kurtosis Hard
A. Both coefficients are one because the distribution has one support point
B. Both coefficients are zero because all central moments vanish
C. is zero while is three by continuity
D. Both coefficients are undefined because their denominators contain zero variance
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