1 $If and are two random variables, the joint cumulative distribution function (CDF) is defined as:$

A.

B.

C.

D.

2 $For a joint CDF, what is the value of ?$

A.

0.5

B.

0

C.

1

D.

Undefined

3 $If the joint probability density function (PDF) of two random variables and is, how is the marginal density function obtained?$

A.

B.

C.

D.

4 $Two random variables and are statistically independent if and only if their joint PDF satisfies:$

A.

B.

C.

D.

5 $The relationship between the joint PDF and the joint CDF is given by:$

A.

B.

C.

D.

6 $Given the conditional density function, the joint density function can be expressed as:$

A.

B.

C.

D.

7 $If and are independent random variables, what is ?$

A.

B.

C.

0

D.

8 $The covariance of two random variables and, denoted as or, is defined as:$

A.

B.

C.

D.

9 $If the correlation coefficient, the random variables and are said to be:$

A.

Mutually Exclusive

B.

Uncorrelated

C.

Statistically Independent

D.

Orthogonal

10 $Two random variables and are said to be orthogonal if:$

A.

B.

C.

D.

11 $What is the probability density function of the sum of two independent random variables and ?$

A.

The convolution of and

B.

C.

D.

12 $The Central Limit Theorem states that the distribution of the sum of a large number of independent, identically distributed (i.i.d.) random variables approaches:$

A.

A Poisson distribution

B.

A Uniform distribution

C.

A Gaussian (Normal) distribution

D.

An Exponential distribution

13 $If for all, then and are:$

A.

Orthogonal

B.

Correlated

C.

Dependent

D.

Independent

14 $For a joint PDF, the volume under the surface over the entire -plane is:$

A.

0

B.

C.

1

D.

Undefined

15 $The joint moment of order about the origin is defined as :$

A.

B.

C.

D.

16 $The central moment corresponds to which statistical property?$

A.

Variance of X

B.

Correlation of X and Y

C.

Mean of XY

D.

Covariance of X and Y

17 $If, then the expected value is equal to:$

A.

B.

C.

D.

18 $The range of the correlation coefficient is:$

A.

B.

C.

D.

19 $If the joint PDF is for and 0 otherwise, what is the value of ?$

A.

4

B.

2

C.

0.5

D.

1

20 $The property yields:$

A.

1

B.

C.

D.

0

21 $Which of the following is true for the variance of the sum of two independent random variables and ?$

A.

B.

C.

D.

22 $If is the joint density, what is ?$

A.

B.

C.

D.

23 $What is the relationship between Correlation and Covariance ?$

A.

B.

C.

D.

24 $The conditional distribution function is defined as:$

A.

B.

C.

D.

25 $If and are jointly Gaussian random variables and they are uncorrelated, then they are:$

A.

Dependent

B.

Independent

C.

Mutually Exclusive

D.

Identical

26 $The joint characteristic function is defined as:$

A.

B.

C.

D.

27 $If and are independent, the joint characteristic function equals:$

A.

B.

0

C.

D.

28 $What is the expected value of a function, denoted ?$

A.

B.

C.

D.

29 $If, the characteristic function of is ?$

A.

(always)

B.

(only if independent)

C.

D.

30 $Which of the following conditions ensures that is a valid joint PDF?$

A.

and

B.

and

C.

D.

is continuous everywhere

31 $The correlation coefficient is calculated as:$

A.

B.

C.

D.

32 $For independent random variables, the mean of the sum is:$

A.

B.

C.

D.

0

33 $For independent random variables, the variance of the sum is:$

A.

The sum of the standard deviations

B.

The sum of variances plus twice covariances

C.

The sum of the variances

D.

The product of the variances

34 $The conditional density is valid only if:$

A.

B.

C.

D.

and are independent

35 $Point conditioning of a CDF involves determining:$

A.

B.

C.

D.

36 $If and are independent, then is equal to:$

A.

1

B.

C.

D.

37 $The Cauchy-Schwarz inequality for expectations states that is:$

A.

B.

C.

D.

38 $Consider the vector random variable . The joint PDF is a function mapping:$

A.

B.

C.

D.

39 $If where and are constants, what is ?$

A.

B.

C.

D.

40 $If and are independent, what is ?$

A.

B.

C.

D.

41 $For joint variables and, if, this implies:$

A.

and are independent

B.

and are orthogonal

C.

and are identical

D.

and are uncorrelated

42 $The second order joint moment about the origin is equivalent to:$

A.

B.

C.

D.

43 $In the context of the Central Limit Theorem, as, the mean of the sample sum (where have mean) is:$

A.

B.

0

C.

D.

44 $What is the joint CDF property ?$

A.

0

B.

1

C.

D.

45 $The skewness of the joint distribution involves joint central moments of order:$

A.

2

B.

1

C.

4

D.

3

46 $Which term describes the 'center of gravity' of the joint PDF mass?$

A.

B.

C.

D.

47 $If and are independent exponential random variables, the sum follows a:$

A.

Gaussian distribution

B.

Rayleigh distribution

C.

Uniform distribution

D.

Gamma distribution (Erlang)

48 $The conditional expectation is a function of:$

A.

B.

Constant

C.

Both and

D.

49 $For discrete random variables and, the joint PMF is . The marginal PMF is:$

A.

B.

C.

D.

50 $If the joint PDF is for, then and are:$

A.

Independent Uniform variables

B.

Dependent Exponential variables

C.

Dependent Gaussian variables

D.

Independent Exponential variables

Unit 4 - Practice Quiz