1 $The Cumulative Distribution Function (CDF),, of a random variable is defined as:$

A.

B.

C.

D.

2 $For a continuous random variable with Probability Density Function (PDF), which of the following conditions must hold true?$

A.

B.

for some

C.

D.

3 $If is the CDF of a continuous random variable, then the PDF is given by:$

A.

B.

C.

D.

4 $For a continuous random variable, what is where is a specific constant?$

A.

$1$

B.

C.

$0$

D.

$0.5$

5 $Which of the following properties is TRUE for any CDF ?$

A.

B.

C.

is a non-increasing function

D.

6 $The probability can be expressed in terms of the CDF as:$

A.

B.

C.

D.

7 $A Binomial random variable with parameters and represents:$

A.

The time between events in a Poisson process.

B.

The number of successes in independent Bernoulli trials.

C.

The magnitude of noise in a communication system.

D.

The number of trials until the first success occurs.

8 $The mean of a Binomial distribution with parameters and is:$

A.

B.

C.

D.

9 $The variance of a Binomial random variable is given by:$

A.

B.

C.

D.

10 $The Poisson distribution is a limiting case of the Binomial distribution when:$

A.

,, and (constant)

B.

,

C.

,, and (constant)

D.

,

11 $The Probability Mass Function (PMF) of a Poisson random variable with parameter is:$

A.

B.

C.

D.

12 $For a Poisson distribution, the relationship between Mean () and Variance () is:$

A.

B.

C.

D.

13 $The PDF of a Uniform random variable defined over the interval is:$

A.

for

B.

for

C.

for

D.

for

14 $What is the mean of a Uniform random variable distributed in ?$

A.

B.

C.

D.

15 $The variance of a Uniform random variable is:$

A.

B.

C.

D.

16 $The Exponential distribution is often used to model:$

A.

The sum of many independent random variables.

B.

The waiting time between independent events occurring at a constant average rate.

C.

The envelope of a narrow-band noise signal.

D.

The number of successes in n trials.

17 $The PDF of an Exponential random variable with parameter is:$

A.

B.

C.

D.

18 $What is the mean of an Exponential random variable with PDF ?$

A.

B.

C.

D.

19 $The "Memoryless Property" is a unique characteristic of which continuous distribution?$

A.

Rayleigh

B.

Uniform

C.

Exponential

D.

Gaussian

20 $Which distribution is commonly known as the "Bell Curve"?$

A.

Rayleigh

B.

Gaussian (Normal)

C.

Exponential

D.

Poisson

21 $The PDF of a Gaussian random variable with mean and variance is:$

A.

B.

C.

D.

22 $For a Standard Normal Random Variable, the parameters are:$

A.

B.

C.

D.

23 $The Gaussian density function is symmetric about:$

A.

B.

C.

D.

24 $The Rayleigh distribution is typically observed in:$

A.

Waiting times at a bus stop.

B.

The number of phone calls in an hour.

C.

The envelope of a narrow-band Gaussian noise.

D.

Outcomes of a coin toss.

25 $The PDF of a Rayleigh random variable for is given by:$

A.

B.

C.

D.

26 $If, the conditional distribution function is defined as:$

A.

B.

C.

D.

27 $The conditional density function is related to the conditional distribution function by:$

A.

B.

C.

D.

28 $If event, what is for ?$

A.

$0$

B.

C.

$1$

D.

29 $If event, what is for ?$

A.

B.

C.

D.

30 $Which of the following properties holds for a conditional density function ?$

A.

can be negative.

B.

always.

C.

D.

31 $If is a Uniform random variable in, what is the value of PDF at ?$

A.

$1$

B.

$0.25$

C.

$0$

D.

$0.5$

32 $For, calculate .$

A.

$0.3$

B.

$0.5$

C.

$0.2$

D.

$0.7$

33 $Find the variance of a random variable with and .$

A.

$4$

B.

$9$

C.

$11$

D.

$15$

34 $If is Gaussian with mean 0 and variance 4, what is the value of the PDF at ?$

A.

B.

C.

D.

$0$

35 $For a Poisson random variable, if, what is the mean?$

A.

$4$

B.

$1$

C.

$2$

D.

36 $Given the conditional density, the probability is calculated as:$

A.

B.

C.

D.

37 $If has an Exponential distribution with mean 2, what is the variance?$

A.

$0.25$

B.

$4$

C.

$2$

D.

$0.5$

38 $Which method is NOT a standard way to define a conditioning event for a continuous random variable ?$

A.

B.

C.

D.

39 $In the Binomial distribution, if and, the distribution is:$

A.

Symmetric

B.

Skewed to the left

C.

Uniform

D.

Skewed to the right

40 $The area under the standard normal curve between and is approximately:$

A.

$0.68$

B.

$0.50$

C.

$0.95$

D.

$0.99$

41 $If is a random variable and is a constant, then is:$

A.

$1$

B.

$0.5$

C.

$0$

D.

42 $Let be Gaussian with . If, what is the value?$

A.

$1$

B.

C.

D.

$0.5$

43 $For a Rayleigh distribution, the mode (peak of the PDF) occurs at:$

A.

B.

C.

D.

44 $The Q-function,, often used with Gaussian variables, is defined as:$

A.

B.

C.

D.

45 $If a density function for and 0 otherwise, find .$

A.

$1$

B.

$0.25$

C.

$2$

D.

$0.5$

46 $Which distribution has the property that the probability of occurrence in a small interval is proportional to the length of the interval (for a process)?$

A.

Gaussian

B.

Binomial

C.

Poisson

D.

Rayleigh

47 $If is even, i.e.,, then the CDF is:$

A.

$0.5$

B.

$0$

C.

$1$

D.

Undefined

48 $The CDF of the Exponential distribution for is:$

A.

B.

C.

D.

49 $Conditioning a Gaussian variable on the event results in:$

A.

A Gaussian distribution.

B.

A Rayleigh distribution.

C.

A one-sided (truncated) Gaussian distribution.

D.

A Uniform distribution.

50 $A fair coin is tossed 3 times. Let be the number of heads. What is ?$

A.

B.

C.

D.

Unit 2 - Practice Quiz