Probability functions and densities, distribution functions, mathematical expectation, marginal and conditional distributions, conditional expectation - One Line Questions

1. What is the range of a probability density function (PDF) for a continuous random variable? — [0, ∞)
2. 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)? — ∫ f(x, y) dx over all x
3. What is the value of F(-∞) for any cumulative distribution function F(x)? — 0
4. 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: — 0
5. What is the value of F(∞) for any cumulative distribution function F(x)? — 1
6. For a discrete random variable X, the sum of probabilities P(X=x) over all possible values of x must equal: — 1
7. Consider two events A and B. If P(A|B) = P(A), what can be concluded about the events? — A and B are independent
8. What does a probability function assign to each outcome in a sample space? — A numerical value representing the likelihood of the outcome
9. In the context of mathematical expectation, what does E(c) represent, where 'c' is a constant? — c
10. What is the conditional expectation E(X|Y=y) if X and Y are independent? — E(X)
11. If X and Y are random variables, what is E(X + Y)? — E(X) + E(Y)
12. If X and Y are independent, what is the conditional expectation E(Y|X=x)? — E(Y)
13. 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: — E(Y) = E[E(Y|X)]
14. Consider the joint PDF f(x, y) of X and Y. The conditional PDF f_{Y|X}(y|x) is proportional to: — f(x, y)
15. 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)? — f(x, y) / f_X(x), provided f_X(x) > 0
16. 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)? — f(x, y) = f_X(x) * f_Y(y)
17. What is the distribution function (also known as the cumulative distribution function or CDF) of a random variable X, denoted by F(x)? — F(x) = P(X ≤ x)
18. For a continuous random variable X, its probability density function f(x) must satisfy two conditions. What are they? — f(x) ≥ 0 for all x, and the integral of f(x) over its entire range is 1
19. Which of these is NOT a property of a cumulative distribution function (CDF), F(x)? — F(x) is always positive
20. Which of the following is a fundamental property of a cumulative distribution function F(x)? — F(x) is non-decreasing
21. What is the concept of conditional expectation used to model? — How the average value of one variable changes when another variable's value is known
22. 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? — Integrate f(x, y) with respect to y over its entire range
23. What is the property E(aX + b) = aE(X) + b known as, where 'a' and 'b' are constants? — Linearity of Expectation
24. The marginal distribution of a random variable describes its probability distribution independently of what? — Other random variables in the system
25. The expected value of a Bernoulli random variable with parameter p is: — p
26. What is the conditional probability of event A occurring given that event B has already occurred, denoted by P(A|B)? — P(A ∩ B) / P(B), provided P(B) > 0
27. The Bayes' Theorem relates conditional probabilities. If we have P(A|B) and P(B), what can we find using Bayes' Theorem? — P(B|A)
28. 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)? — P(X=x, Y=y) / P(X=x), provided P(X=x) > 0
29. 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? — P(X=x) ≥ 0 and Σ P(X=x) = 1
30. When X and Y are independent, what is the conditional probability P(Y=y | X=x)? — P(Y=y)
31. What is the primary difference between a probability mass function (PMF) and a probability density function (PDF)? — All of the above
32. Which function is used to define the probability distribution for continuous random variables? — Probability Density Function (PDF)
33. 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)? — Sum of P(X=x, Y=y) over all possible values of y
34. The marginal distribution of X can be obtained from the joint distribution P(X=x, Y=y) by: — Summing over all possible values of Y
35. The expectation of the sum of two random variables is equal to the sum of their expectations. This is a statement of: — The Additivity of Expectation
36. For continuous random variables, the conditional expectation E(Y|X=x) is calculated using which function? — The conditional probability density function f_{Y|X}(y|x)
37. What is the mathematical expectation of a constant random variable? — The constant itself
38. A key property of conditional expectation is E[E(Y|X)] = E(Y). What does this property signify? — The expected value of the conditional expectation of Y given X is equal to the overall expected value of Y
39. What is the conditional expectation of Y given X=x, denoted by E(Y|X=x), for discrete random variables? — The expected value of Y calculated using the conditional probability mass function P(Y=y | X=x)
40. If X is a continuous random variable with PDF f(x), how is its mathematical expectation E(X) calculated? — The integral of x * f(x) over the entire range of X
41. What does the mathematical expectation of a random variable represent? — The long-run average value of the random variable
42. In a joint distribution of X and Y, P(X=x | Y=y) represents what? — The probability of X taking value 'x' given that Y has taken value 'y'
43. What does the conditional distribution P(X=x | Y=y) describe? — The probability of X=x for a fixed value of Y=y
44. What is the domain of a probability function for a discrete random variable? — The set of all possible outcomes
45. What is the mathematical expectation of a discrete random variable X, denoted by E(X)? — The sum of each possible value multiplied by its probability
46. The cumulative distribution function F(x) is always a function of what variable? — The upper limit of the probability range
47. If P(A ∩ B) = P(A) * P(B), what is the relationship between events A and B? — They are independent
48. What is the primary purpose of a probability density function (PDF) for a continuous random variable? — To describe the relative likelihood for a continuous random variable to take on a given value
49. The integral of the cumulative distribution function F(x) from -∞ to ∞ must be: — Undefined (as F(x) is not a PDF)
50. If X is a discrete random variable and g(X) is a function of X, what is E[g(X)]? — Σ g(x) P(X=x) over all x
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