TRB BT Assistant - Probability Theory and Random Variables One Line Questions & Answers

1. When flipping a fair coin, what is the probability of getting heads? — 0.5

2. What is the probability of an impossible event? — 0

3. What is the probability of a certain event? — 1

4. For a continuous random variable, what is the probability of it taking on a specific single value? — 0

5. When rolling a fair six-sided die, what is the probability of rolling a 4? — 1/6

6. What is a stochastic process? — A collection of random variables indexed by time or space.

7. A sample space is defined as: — The set of all possible outcomes of an experiment.

8. What is a Markov chain? — A stochastic process where the future state depends only on the present state, not on the past states.

9. What is a random variable? —

10. Which distribution is often called the 'bell curve' and is characterized by its mean and variance? — Normal Distribution

11. What is the probability mass function (PMF) used for? — Discrete random variables.

12. What is the correlation coefficient, ρ(X, Y)? — Cov(X, Y) / (σ_X * σ_Y)

13. What type of random variable can take on any value within a given range? — Continuous Random Variable

14. What type of random variable can only take on a finite or countably infinite number of values? — Discrete Random Variable

15. What is the probability density function (PDF) used for? — Continuous random variables.

16. What is the covariance of two random variables X and Y, Cov(X, Y)? — E[(X - E[X])(Y - E[Y])]

17. What is the cumulative distribution function (CDF), F(x), for a random variable X? — F(x) = P(X <= x)

18. Which property of random variables states that E[aX + b] = aE[X] + b for constants a and b? — Linearity of Expectation

19. What is the key characteristic of a Bernoulli random variable? — It has only two possible outcomes: success (1) or failure (0).

20. Which theorem is fundamental to statistical inference and allows us to use normal distributions to approximate other distributions? — Central Limit Theorem

21. What are the parameters of a Normal Distribution? — mean (μ) and standard deviation (σ)

22. What is the conditional probability of event A given that event B has occurred, P(A|B)? — P(A ∩ B) / P(B)

23. If events A and B are independent, which relationship holds true? — P(A and B) = P(A) * P(B)

24. What is the probability of the union of two events A and B, P(A U B)? — P(A) + P(B) - P(A ∩ B)

25. If P(A) denotes the probability of event A, what is the range of possible values for P(A)? — 0 <= P(A) <= 1

26. Which probability distribution is used to model the number of successes in a fixed number of independent Bernoulli trials? — Binomial Distribution

27. What are marginal probabilities in the context of joint distributions? — Probabilities of individual random variables, obtained by summing or integrating the joint distribution.

28. How is the expected value of a discrete random variable calculated? — Sum of (value * probability of value).

29. What is the expected value of a random variable X, denoted E[X]? — The average value of X over an infinite number of trials.

30. What is the mode of a probability distribution? — The value with the highest probability (for discrete) or highest density (for continuous).

31. What does the variance of a random variable measure? — The spread or dispersion of the values around the expected value.

32. Which of the following is an example of a discrete random variable? — The number of heads in 10 coin flips.

33. What is the median of a probability distribution? — The value that divides the probability distribution into two equal halves.

34. Which of the following is an example of a continuous random variable? — The speed of a car.

35. The Poisson distribution is typically used to model: — The number of events occurring in a fixed interval of time or space.

36. The Exponential distribution is commonly used to model: — The time between events in a Poisson process.

37. If X ~ Poi(λ), what does 'λ' represent in the Poisson distribution? — The average rate of occurrence.

38. What is a joint probability distribution? — The probability of two or more random variables occurring simultaneously.

39. What does the Law of Large Numbers state? — As the number of trials increases, the sample average converges to the expected value.

40. Which of the following is NOT a basic axiom of probability (Kolmogorov's axioms)? — The probability of an event is directly proportional to the number of favorable outcomes.

41. If X ~ B(n, p), what does 'n' represent in the Binomial distribution? — The number of trials.

42. An event is defined as: — A subset of the sample space.

43. The standard deviation of a random variable is: — The square root of the variance.

44. What is the fundamental concept of probability theory? — The quantification of uncertainty or chance.

45. What is the Central Limit Theorem (CLT)? — The distribution of the sample mean of sufficiently large samples will be approximately normally distributed, regardless of the population distribution.

46. In the context of random variables, what does it mean for X and Y to be identically distributed? — They have the same probability distribution.

47. If Cov(X, Y) = 0, what can we say about X and Y? — They are uncorrelated.

48. What does it mean for two events A and B to be mutually exclusive? — The occurrence of one event prevents the occurrence of the other.

49. What is the primary purpose of a probability distribution? — To describe the likelihood of different possible outcomes for a random variable.

50. For independent random variables X and Y, what is Var(X + Y)? — Var(X) + Var(Y)

Probability Theory and Random Variables - One Line Questions