TRB PG Assistant - Probability functions and densities, distribution functions, mathematical expectation, marginal and conditional distributions, conditional expectation Question Bank with Options

1. Consider the joint PDF f(x, y) of X and Y. The conditional PDF f_{Y|X}(y|x) is proportional to:

A) f(x, y)

B) f_X(x)

C) f_Y(y)

D) f(x, y) / f_X(x)

2. The expectation of the sum of two random variables is equal to the sum of their expectations. This is a statement of:

A) The Additivity of Expectation

B) The Multiplicativity of Expectation

C) The Linearity of Expectation

D) The Law of Large Numbers

3. What is the primary difference between a probability mass function (PMF) and a probability density function (PDF)?

A) PMF applies to discrete variables, PDF applies to continuous variables

B) PMF values sum to 1, PDF values integrate to 1

C) PMF gives probability at a point, PDF gives density at a point

D) All of the above

4. If P(A ∩ B) = P(A) * P(B), what is the relationship between events A and B?

A) They are independent

B) They are mutually exclusive

C) They are dependent

D) They are complementary

5. The marginal distribution of X can be obtained from the joint distribution P(X=x, Y=y) by:

A) Summing over all possible values of Y

B) Summing over all possible values of X

C) Multiplying by the marginal of Y

D) Dividing by the marginal of Y

6. What is the conditional expectation E(X|Y=y) if X and Y are independent?

A) E(X)

B) E(Y)

C) E(X) * E(Y)

D) 0

7. If X is a discrete random variable and g(X) is a function of X, what is E[g(X)]?

A) Σ g(x) P(X=x) over all x

B) Σ x P(X=x) over all x

C) Σ P(X=x) over all x

D) Σ g(x) over all x

8. The integral of the cumulative distribution function F(x) from -∞ to ∞ must be:

A) Undefined (as F(x) is not a PDF)

B) 1

C) 0

D) Dependent on the specific distribution

9. Which function is used to define the probability distribution for continuous random variables?

A) Probability Density Function (PDF)

B) Probability Mass Function (PMF)

C) Cumulative Distribution Function (CDF)

D) Probability Generating Function (PGF)

10. What is the mathematical expectation of a constant random variable?

A) The constant itself

B) Zero

C) One

D) Undefined

11. Consider two events A and B. If P(A|B) = P(A), what can be concluded about the events?

A) A and B are independent

B) A and B are mutually exclusive

C) A and B are dependent

D) A is a subset of B

12. The law of total expectation states that E(Y) can be computed by averaging the conditional expectation of Y given X over all possible values of X. This is represented as:

A) E(Y) = E[E(Y|X)]

B) E(Y) = E[E(X|Y)]

C) E(Y) = E(X) * E(Y)

D) E(Y) = E(X) + E(Y)

13. What does the conditional distribution P(X=x | Y=y) describe?

A) The probability of X=x for a fixed value of Y=y

B) The probability of Y=y for a fixed value of X=x

C) The joint probability of X=x and Y=y

D) The marginal probability of X=x

14. If f(x, y) is the joint PDF of continuous random variables X and Y, what is the marginal PDF of Y, f_Y(y)?

A) ∫ f(x, y) dx over all x

B) ∫ f(x, y) dy over all y

C) ∫ f(x, y) dx dy

D) f(x, y)

15. For a discrete random variable X, the sum of probabilities P(X=x) over all possible values of x must equal:

A) 1

B) 0

C) The expected value of X

D) The variance of X

16. The expected value of a Bernoulli random variable with parameter p is:

A) p

B) 1-p

C) p^2

D) 1

17. Which of these is NOT a property of a cumulative distribution function (CDF), F(x)?

A) F(x) is always positive

B) lim_{x→-∞} F(x) = 0

C) lim_{x→∞} F(x) = 1

D) F(x) is non-decreasing

18. The definition of a probability density function (PDF) f(x) for a continuous random variable X implies that P(X=a) for any specific value 'a' is:

A) 0

B) f(a)

C) 1

D) Undefined

19. What is the concept of conditional expectation used to model?

A) How the average value of one variable changes when another variable's value is known

B) The total probability of a system

C) The independence of two variables

D) The maximum possible value of a variable

20. The Bayes' Theorem relates conditional probabilities. If we have P(A|B) and P(B), what can we find using Bayes' Theorem?

A) P(B|A)

B) P(A ∩ B)

C) P(A)

D) P(B)

21. In a joint distribution of X and Y, P(X=x | Y=y) represents what?

A) The probability of X taking value 'x' given that Y has taken value 'y'

B) The probability of Y taking value 'y' given that X has taken value 'x'

C) The joint probability of X=x and Y=y

D) The marginal probability of X=x

22. The marginal distribution of a random variable describes its probability distribution independently of what?

A) Other random variables in the system

B) Its own expected value

C) Its own variance

D) The constant '0'

23. If X and Y are random variables, what is E(X + Y)?

A) E(X) + E(Y)

B) E(X) * E(Y)

C) E(X) - E(Y)

D) E(X) / E(Y)

24. What is the property E(aX + b) = aE(X) + b known as, where 'a' and 'b' are constants?

A) Linearity of Expectation

B) Additivity of Expectation

C) Multiplicativity of Expectation

D) Homoscedasticity

25. In the context of mathematical expectation, what does E(c) represent, where 'c' is a constant?

A) c

B) 0

C) 1

D) E(X)

26. What is the value of F(∞) for any cumulative distribution function F(x)?

A) 1

B) 0

C) 0.5

D) Undefined

27. What is the value of F(-∞) for any cumulative distribution function F(x)?

A) 0

B) 1

C) 0.5

D) Undefined

28. The cumulative distribution function F(x) is always a function of what variable?

A) The upper limit of the probability range

B) The probability value

C) The expected value

D) The variance

29. What is the range of a probability density function (PDF) for a continuous random variable?

A) [0, ∞)

B) [0, 1]

C) (-∞, ∞)

D) {0, 1}

30. What is the domain of a probability function for a discrete random variable?

A) The set of all possible outcomes

B) The set of all real numbers

C) A single specific value

D) The interval [0, 1]

31. If X and Y are independent, what is the conditional expectation E(Y|X=x)?

A) E(Y)

B) E(X)

C) E(X) * E(Y)

D) 0

32. When X and Y are independent, what is the conditional probability P(Y=y | X=x)?

A) P(Y=y)

B) P(X=x)

C) P(X=x) * P(Y=y)

D) 1

33. If X and Y are independent random variables, what is the relationship between their joint PDF f(x, y) and marginal PDFs f_X(x) and f_Y(y)?

A) f(x, y) = f_X(x) * f_Y(y)

B) f(x, y) = f_X(x) + f_Y(y)

C) f(x, y) = f_X(x) / f_Y(y)

D) f(x, y) = 1

34. A key property of conditional expectation is E[E(Y|X)] = E(Y). What does this property signify?

A) The expected value of the conditional expectation of Y given X is equal to the overall expected value of Y

B) The conditional expectation of Y given X is always equal to the marginal expectation of Y

C) The expected value of Y given X is always equal to the expected value of X given Y

D) The expected value of Y is always zero

35. For continuous random variables, the conditional expectation E(Y|X=x) is calculated using which function?

A) The conditional probability density function f_{Y|X}(y|x)

B) The joint probability density function f(x, y)

C) The marginal probability density function f_X(x)

D) The cumulative distribution function F(y|x)

36. What is the conditional expectation of Y given X=x, denoted by E(Y|X=x), for discrete random variables?

A) The expected value of Y calculated using the conditional probability mass function P(Y=y | X=x)

B) The expected value of Y calculated using the marginal probability mass function P(Y=y)

C) The expected value of X calculated using the conditional probability mass function P(Y=y | X=x)

D) The sum of P(Y=y | X=x) over all y

37. If X and Y are continuous random variables with joint PDF f(x, y), what is the conditional PDF of Y given X=x, denoted by f_{Y|X}(y|x)?

A) f(x, y) / f_X(x), provided f_X(x) > 0

B) f(x, y) * f_X(x)

C) f_X(x) / f_{Y|X}(y|x)

D) f(x, y) / f_Y(y)

38. For two discrete random variables X and Y, what is the conditional probability mass function of Y given X=x, denoted by P(Y=y | X=x)?

A) P(X=x, Y=y) / P(X=x), provided P(X=x) > 0

B) P(X=x, Y=y) * P(X=x)

C) P(Y=y) / P(X=x)

D) P(X=x) / P(Y=y)

39. What is the conditional probability of event A occurring given that event B has already occurred, denoted by P(A|B)?

A) P(A ∩ B) / P(B), provided P(B) > 0

B) P(A ∩ B) * P(B)

C) P(A) / P(B)

D) P(A) + P(B)

40. For two continuous random variables X and Y with joint PDF f(x, y), how is the marginal PDF of X, denoted by f_X(x), obtained?

A) Integrate f(x, y) with respect to y over its entire range

B) Integrate f(x, y) with respect to x over its entire range

C) Integrate f(x, y) with respect to both x and y

D) Find the maximum value of f(x, y) for a fixed x

41. Consider a joint probability distribution function P(X=x, Y=y) for two discrete random variables X and Y. What is the marginal probability mass function of X, denoted by P(X=x)?

A) Sum of P(X=x, Y=y) over all possible values of y

B) Sum of P(X=x, Y=y) over all possible values of x

C) The maximum value of P(X=x, Y=y) for a fixed x

D) The product of P(X=x) and P(Y=y)

42. What does the mathematical expectation of a random variable represent?

A) The long-run average value of the random variable

B) The most frequent value of the random variable

C) The spread or dispersion of the random variable's values

D) The probability of achieving the average value

43. If X is a continuous random variable with PDF f(x), how is its mathematical expectation E(X) calculated?

A) The integral of x * f(x) over the entire range of X

B) The integral of f(x) over the entire range of X

C) The integral of x * F(x) over the entire range of X

D) The value of f(x) where f(x) is maximum

44. What is the mathematical expectation of a discrete random variable X, denoted by E(X)?

A) The sum of each possible value multiplied by its probability

B) The probability of the most likely outcome

C) The sum of all possible values

D) The variance of the distribution

45. Which of the following is a fundamental property of a cumulative distribution function F(x)?

A) F(x) is non-decreasing

B) F(x) is always decreasing

C) F(x) is constant

D) F(x) is always negative

46. What is the distribution function (also known as the cumulative distribution function or CDF) of a random variable X, denoted by F(x)?

A) F(x) = P(X ≤ x)

B) F(x) = P(X < x)

C) F(x) = P(X = x)

D) F(x) = P(X > x)

47. For a continuous random variable X, its probability density function f(x) must satisfy two conditions. What are they?

A) f(x) ≥ 0 for all x, and the integral of f(x) over its entire range is 1

B) f(x) > 0 for all x, and the integral of f(x) over its entire range is 0

C) f(x) ≥ 0 for all x, and the integral of f(x) over its entire range is less than 1

D) f(x) = 1 for all x, and the integral of f(x) over its entire range is 1

48. What is the primary purpose of a probability density function (PDF) for a continuous random variable?

A) To describe the relative likelihood for a continuous random variable to take on a given value

B) To assign a specific probability to each exact value

C) To represent the cumulative probability up to a certain value

D) To define the expected value of the variable

49. For a discrete random variable X, the probability mass function (PMF) is denoted by P(X=x). What property must P(X=x) satisfy for all x?

A) P(X=x) ≥ 0 and Σ P(X=x) = 1

B) P(X=x) > 0 and Σ P(X=x) < 1

C) P(X=x) ≤ 0 and Σ P(X=x) = 0

D) P(X=x) = 1 for all x

50. What does a probability function assign to each outcome in a sample space?

A) A numerical value representing the likelihood of the outcome

B) A set of all possible outcomes

C) The expected value of the outcome

D) The variance of the outcome

Probability functions and densities, distribution functions, mathematical expectation, marginal and conditional distributions, conditional expectation - Question Bank