For a continuous random variable $X$ with probability density function $f(x)$, which expression gives $E[X]$?

$\int_{-\infty}^{\infty} x f(x)\,dx$

$\int_{-\infty}^{\infty} f(x)\,dx$

$\int_{-\infty}^{\infty} x\,dx$

$\int_{-\infty}^{\infty} x^2 f(x)\,dx$

For a continuous random variable $X$, how is $E[g(X)]$ calculated from its density $f(x)$?

$\int_{-\infty}^{\infty} g(x)f(x)\,dx$

$\int_{-\infty}^{\infty} g(x)\,dx$

$\int_{-\infty}^{\infty} xf(x)\,dx$

$\int_{-\infty}^{\infty} f(g(x))\,dx$

ECAP790 Unit 3: Mathematical Expectations - MCQ Practice | MCA - ODL

1 $What does the mathematical expectation of a random variable represent?$

Definition of mathematical expectation Easy

A.

Its most frequent value

B.

Its smallest possible value

C.

Its largest possible value

D.

Its probability-weighted average value

2 $A random variable takes values and with probabilities and . What is ?$

Expected value of a random variable Easy

A.

B.

C.

D.

3 $For a continuous random variable with probability density function, which expression gives ?$

Expected value of a random variable Easy

A.

B.

C.

D.

4 $If takes values and with equal probability, what is ?$

Expected value of a function of a random variable Easy

A.

B.

C.

D.

5 $For a continuous random variable, how is calculated from its density ?$

Expected value of a function of a random variable Easy

A.

B.

C.

D.

6 $If, what is ?$

Properties of expectations Easy

A.

B.

C.

D.

7 $If is a constant, what is ?$

Properties of expectations Easy

A.

B.

C.

D.

8 $Which measure of central tendency is the arithmetic average of observed values?$

Measures of Central Tendency Easy

A.

Median

B.

Range

C.

Mean

D.

Mode

9 $For a continuous distribution, the median satisfies which condition?$

Measures of Central Tendency Easy

A.

B.

C.

D.

10 $For a continuous probability distribution, the mode is the value at which which function is greatest?$

Measures of Central Tendency Easy

A.

Expectation function

B.

Variance function

C.

Cumulative distribution function

D.

Probability density function

11 $Which measure of dispersion equals the largest value minus the smallest value?$

Dispersion Easy

A.

Mean

B.

Range

C.

Median

D.

Mode

12 $Which measure of dispersion is the positive square root of variance?$

Dispersion Easy

A.

Quartile deviation

B.

Standard deviation

C.

Coefficient of skewness

D.

Mean deviation

13 $What is the skewness of a perfectly symmetric continuous distribution?$

Skewness and Kurtosis for continuous probability distribution Easy

A.

B.

C.

D.

14 $A continuous distribution with a long tail extending to the right is described as:$

Skewness and Kurtosis for continuous probability distribution Easy

A.

Negatively skewed

B.

Uniformly distributed

C.

Perfectly symmetric

D.

Positively skewed

15 $A normal distribution is commonly classified as which type based on kurtosis?$

Skewness and Kurtosis for continuous probability distribution Easy

A.

Mesokurtic

B.

Platykurtic

C.

Leptokurtic

D.

Asymmetric

16 $For a continuous random variable, what does the cumulative distribution function represent?$

Continuous distribution function Easy

A.

only

B.

only

C.

only

D.

17 $Which statement is always true for a cumulative distribution function ?$

Continuous distribution function Easy

A.

It is always negative

B.

It is non-decreasing

C.

It is non-increasing

D.

It is always constant

18 $Which formula expresses the variance of a random variable ?$

Variance Easy

A.

B.

C.

D.

19 $If, what is ?$

Properties of variance Easy

A.

B.

C.

D.

20 $What does a positive covariance between two random variables generally indicate?$

Covariance Easy

A.

They both have zero variance

B.

They always have equal values

C.

They tend to move oppositely

D.

They tend to increase together

21 $A continuous random variable has density for and otherwise. Using the definition of mathematical expectation, find .$

Definition of mathematical expectation Medium

A.

B.

C.

D.

22 $A random variable takes values,, and with probabilities,, and, respectively. What is ?$

Expected value of a random variable Medium

A.

B.

C.

D.

23 $A random variable takes values,, and with probabilities,, and . Find .$

Expected value of a function of a random variable Medium

A.

B.

C.

D.

24 $If and, find .$

Properties of expectations Medium

A.

B.

C.

D.

25 $If has a finite mean, what is ?$

Properties of expectations Medium

A.

B.

C.

D.

26 $A factory uses machine A for of its items and machine B for . Their mean outputs per hour are and items, respectively. What is the overall expected output per hour?$

Properties of expectations Medium

A.

items

B.

items

C.

items

D.

items

27 $A continuous random variable has density for . What is its mean?$

Measures of Central Tendency Medium

A.

B.

C.

D.

28 $The distribution function of is for . What is the median of ?$

Measures of Central Tendency Medium

A.

B.

C.

D.

29 $For the density on, what is the mode?$

Measures of Central Tendency Medium

A.

B.

C.

D.

30 $A continuous random variable is uniformly distributed on . What is its variance?$

Dispersion, Skewness and Kurtosis for continuous probability distribution Medium

A.

B.

C.

D.

31 $A continuous distribution has mean, median, and standard deviation . What is Pearson's second coefficient of skewness?$

Dispersion, Skewness and Kurtosis for continuous probability distribution Medium

A.

B.

C.

D.

32 $For a continuous distribution, the central moments are and . What are its coefficient and type of kurtosis?$

Dispersion, Skewness and Kurtosis for continuous probability distribution Medium

A.

, leptokurtic

B.

, leptokurtic

C.

, platykurtic

D.

, mesokurtic

33 $A random variable has density for and otherwise. Which expression gives its distribution function for ?$

Continuous distribution function Medium

A.

B.

C.

D.

34 $For, a random variable has distribution function . Find .$

Continuous distribution function Medium

A.

B.

C.

D.

35 $If and, what is ?$

Variance Medium

A.

B.

C.

D.

36 $A random variable has and . Find .$

Variance Medium

A.

B.

C.

D.

37 $If, what is ?$

Properties of variance Medium

A.

B.

C.

D.

38 $Suppose,, and . Find .$

Properties of variance Medium

A.

B.

C.

D.

39 $If,, and, what is ?$

Covariance Medium

A.

B.

C.

D.

40 $If, find .$

Covariance Medium

A.

B.

C.

D.

41 $Let have the standard Cauchy density for . What is ?$

Definition of mathematical expectation Hard

A.

because the density is symmetric about zero.

B.

Its Cauchy principal value does not exist because the tails are symmetric.

C.

because the density integrates to one.

D.

is undefined because its positive and negative parts are both infinite.

42 $A continuous random variable has density for and otherwise. What is ?$

Expected value of a random variable Hard

A.

B.

C.

D.

43 $If, find .$

Expected value of a function of a random variable Hard

A.

B.

C.

D.

44 $Suppose and are integrable,,, and . Determine .$

Properties of expectations Hard

A.

B.

C.

D.

45 $A random variable has density for . Which ordered triple gives its mean, median, and mode, respectively?$

Measures of Central Tendency Hard

A.

B.

C.

D.

46 $Let and define . Which ordered triple gives the mean, median, and mode of, respectively?$

Measures of Central Tendency Hard

A.

B.

C.

D.

47 $A continuous random variable has mean and variance . Without further distributional information, what lower bound does Chebyshev's inequality give for ?$

Dispersion Hard

A.

B.

C.

D.

48 $A continuous random variable has standardized skewness and kurtosis . If, what are the standardized skewness and kurtosis of ?$

Skewness and Kurtosis for continuous probability distribution Hard

A.

and

B.

and

C.

and

D.

and

49 $A random variable has distribution function for and for . Find .$

Continuous distribution function Hard

A.

B.

C.

D.

50 $Let for and for . Which statement about the corresponding random variable is correct?$

Continuous distribution function Hard

A.

is undefined due to cancellation of two infinite parts.

B.

because the survival integral diverges.

C.

because the density is rational.

D.

because the distribution has unit scale.

51 $Suppose and . For which value of is minimized, and what is the minimum?$

Variance Hard

A.

and the minimum is

B.

and the minimum is

C.

and the minimum is

D.

and the minimum is

52 $If,, and, what is ?$

Variance Hard

A.

B.

C.

D.

53 $Random variables and satisfy and, but their correlation is unspecified. What is the full possible range of ?$

Properties of variance Hard

A.

B.

C.

D.

54 $Suppose and . What is ?$

Properties of variance Hard

A.

B.

C.

D.

55 $If,, and, find .$

Covariance Hard

A.

B.

C.

D.

56 $Let and be independent and identically distributed random variables with finite, nonzero variance. Define and . Which statement is always true?$

Covariance Hard

A.

and have covariance equal to .

B.

and are independent for every common distribution.

C.

and are uncorrelated but need not be independent.

D.

and are negatively correlated for every common distribution.

57 $If has an exponential distribution with rate, determine .$

Expected value of a function of a random variable Hard

A.

B.

C.

D.

58 $A random variable has density for . What are its standardized skewness and excess kurtosis?$

Skewness and Kurtosis for continuous probability distribution Hard

A.

and

B.

and

C.

and

D.

and

59 $For events and, suppose,, and . Find .$

Covariance Hard

A.

B.

C.

D.

60 $A nonnegative random variable has survival function for . What is ?$

Expected value of a random variable Hard

A.

B.

C.

D.

Unit 3: Mathematical Expectations - Practice Quiz

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