1 $Which of the following defines the sample space of an experiment?$

A.

The set of all possible outcomes.

B.

The set of outcomes satisfying a specific condition.

C.

The set of all successful outcomes.

D.

The numerical value assigned to an outcome.

2 $If set and set are disjoint (mutually exclusive), which of the following is true regarding their intersection?$

A.

B.

C.

D.

3 $According to De Morgan’s Laws, the complement of the union of two sets is equal to:$

A.

B.

C.

D.

4 $A sample space is called discrete if:$

A.

The probabilities of outcomes are unknown.

B.

It contains uncountable outcomes.

C.

It contains a finite or countably infinite number of outcomes.

D.

It consists of an interval of real numbers.

5 $Which of the following represents the classical definition of probability for an event, where is the total number of equally likely outcomes and is the number of favorable outcomes?$

A.

B.

C.

D.

6 $Which of the following is NOT a fundamental axiom of probability?$

A.

If, then .

B.

for any event .

C.

for any two events and .

D.

, where is the sample space.

7 $For any event, what is the relationship between and (the complement of A)?$

A.

B.

C.

D.

8 $The conditional probability of event given event (where) is defined as:$

A.

B.

C.

D.

9 $Two events and are said to be statistically independent if:$

A.

B.

C.

D.

10 $If and are mutually exclusive events with and, what is ?$

A.

0.0

B.

0.1

C.

0.12

D.

0.7

11 $In a mathematical model of an experiment, what is the event usually defined as?$

A.

A random variable.

B.

A subset of the sample space.

C.

A single outcome.

D.

The sample space itself.

12 $Bayes' Theorem relates the conditional probabilities and . The formula is:$

A.

B.

C.

D.

13 $The Total Probability Theorem states that if events form a partition of sample space, then for any event :$

A.

B.

Both A and B

C.

D.

14 $In Bernoulli trials, which of the following conditions must hold?$

A.

The probability of success changes with every trial.

B.

There are only two possible outcomes (Success/Failure) in each trial.

C.

The experiment is performed only once.

D.

The trials are dependent on each other.

15 $If a fair coin is tossed 3 times (Bernoulli trials), what is the probability of getting exactly 2 heads?$

A.

B.

C.

D.

16 $What is the definition of a Random Variable ?$

A.

A function that maps the sample space to the real line .

B.

A subset of the sample space.

C.

A variable that takes random values.

D.

A function that maps real numbers to probabilities.

17 $Which of the following is a necessary condition for a function to be a valid random variable?$

A.

.

B.

.

C.

The set must be an event (measurable) for every real .

D.

The function must be continuous everywhere.

18 $A random variable is called discrete if:$

A.

Its range is an uncountably infinite set.

B.

It has a probability density function (PDF).

C.

Its Cumulative Distribution Function (CDF) is continuous everywhere.

D.

Its range is a countable set of values.

19 $A random variable is called continuous if:$

A.

It is defined only for integers.

B.

It can take any value within a specific interval on the real line.

C.

for some constant .

D.

Its CDF,, is a step function.

20 $What characterizes a Mixed Random Variable ?$

A.

Its PDF is zero everywhere.

B.

It is the product of a discrete and a continuous random variable.

C.

Its CDF has both jump discontinuities and continuous increasing segments.

D.

It is the sum of two discrete random variables.

21 $The Cumulative Distribution Function (CDF) of a random variable is defined as:$

A.

B.

C.

D.

22 $Which of the following is NOT a property of a Cumulative Distribution Function (CDF), ?$

A.

and .

B.

C.

is a non-decreasing function.

D.

is always continuous.

23 $For a continuous random variable, the probability of a specific point outcome is:$

A.

Undefined

B.

C.

0

D.

1

24 $If is the Probability Density Function (PDF) of a continuous random variable, then:$

A.

B.

must be .

C.

D.

represents the probability .

25 $How is the Probability Density Function (PDF) obtained from the CDF for a continuous random variable?$

A.

By subtraction:

B.

They are unrelated.

C.

By differentiation:

D.

By integration:

26 $For a discrete random variable, the function is called the:$

A.

Cumulative Mass Function

B.

Probability Density Function (PDF)

C.

Unit Step Function

D.

Probability Mass Function (PMF)

27 $If, then which of the following is true regarding their probabilities?$

A.

B.

C.

D.

28 $The joint probability is equivalent to:$

A.

B.

C.

D.

29 $In a continuous sample space, an event with probability zero :$

A.

Can occur, but is statistically negligible (e.g., a specific point).

B.

Never occurs.

C.

Is the empty set .

D.

Implies the experiment is invalid.

30 $If,, and, then events and are:$

A.

Dependent

B.

Complementary

C.

Independent

D.

Mutually Exclusive

31 $What is the Probability Mass Function (PMF) of a Bernoulli random variable with probability of success ?$

A.

B.

C.

D.

for

32 $For a random variable, the expression in terms of CDF is:$

A.

B.

C.

D.

33 $The Dirac delta function is often used in the PDF of which type of random variable?$

A.

Mixed or Discrete (when represented in continuous notation)

B.

Purely Continuous

C.

Bernoulli only

D.

Gaussian only

34 $A function can be a valid CDF if and only if:$

A.

B.

It is continuous.

C.

It is non-decreasing, right-continuous, .

D.

It is a decreasing function.

35 $Which set operation corresponds to the logical 'OR' operator for events?$

A.

Complement ()

B.

Intersection ()

C.

Union ()

D.

Difference ()

36 $The Law of Total Probability is useful for:$

A.

Calculating the mean of a random variable.

B.

Finding the median of a PDF.

C.

Finding the probability of an event based on a partition of the sample space.

D.

Determining if two events are independent.

37 $Given and, what is ?$

A.

1.3

B.

0.625

C.

0.4

D.

0.3

38 $A partition of the sample space is a collection of events such that:$

A.

They all have equal probability.

B.

They are independent.

C.

Their intersection is .

D.

They are pairwise disjoint and their union is .

39 $In the context of random variables, what does the notation imply?$

A.

is a function of the outcome .

B.

is multiplied by .

C.

is independent of .

D.

is the derivative of .

40 $If is a discrete random variable taking values $1, 2, 3$ with probabilities $0.2, 0.5, 0.3$, what is ?$

A.

0.3

B.

0.7

C.

0.2

D.

0.5

41 $Which of the following is an example of a continuous random variable?$

A.

The value of a roll of a die.

B.

The time until a light bulb burns out.

C.

The number of heads in 10 coin tosses.

D.

The number of defective items in a batch.

42 $If, and are mutually exclusive, what is ?$

A.

0

B.

0.4

C.

0.16

D.

1

43 $What is the value of for a discrete random variable?$

A.

B.

1

C.

0

D.

0.5

44 $In the experiment of tossing two fair coins, let be the number of heads. The range of is:$

A.

B.

C.

D.

45 $Relative frequency probability definition is strictly valid when:$

A.

The outcomes are not equally likely.

B.

We cannot perform an experiment.

C.

The number of trials approaches infinity.

D.

The number of trials is small.

46 $If and, then is:$

A.

A subset of A.

B.

The complement of A ().

C.

Equal to A.

D.

An impossible event.

47 $For a continuous PDF, the dimensionality of is:$

A.

Probability (unitless).

B.

Same as the random variable's unit.

C.

Undefined.

D.

Inverse of the random variable's unit.

48 $Which axiom prevents negative probabilities?$

A.

Independence axiom.

B.

Additivity axiom.

C.

Non-negativity axiom.

D.

Normalization axiom.

49 $If, then:$

A.

and are independent.

B.

and are mutually exclusive.

C.

.

D.

.

50 $Given a CDF, the probability is:$

A.

B.

C.

D.

Unit 1 - Practice Quiz