Quiz1

1. Why do we study probability?

• • a) To understand deterministic systems.

• • b) Because many phenomena do not follow deterministic rules.

• • c) To replace statistics in empirical analysis.

• • d) To ensure all outcomes are predictable.

What is a random variable?

• • a) A function that assigns a real number to each outcome of an experiment.

• • b) A value that remains fixed for every trial of an experiment.

• • c) The same as a deterministic function.

• • d) A rule that eliminates uncertainty.

What is the sum of all probabilities in a probability density function (PDF)?

• • a) Less than 1

• • b) Equal to 1

• • c) Greater than 1

• • d) Depends on the number of outcomes

If two random variables X and Y are independent, what is true?

• • a) P(X = x, Y = y) = P(X = x) + P(Y = y)

• • b) P(X = x, Y = y) = P(X = x) P(Y = y)

• • c) X and Y must have equal variance.

• • d) The covariance between X and Y is always positive.

The expected value of a discrete random variable is:

• • a) The value that occurs most frequently.

• • b) The sum of all possible values.

• • c) The probability-weighted average of all possible values.

• • d) The maximum possible value of the variable.

What is the standard deviation?

• • a) The square root of the variance.

• • b) The sum of squared deviations from the mean.

• • c) The measure of central tendency.

• • d) Always equal to the variance.

Which of the following statements about the normal distribution is true?

• • a) It is always skewed to the right.

• • b) The mean, median, and mode are the same.

• • c) The probability of values far from the mean is always zero.

• • d) It only applies to discrete variables.

The correlation coefficient between two random variables X and Y is given by:

• • a) Cov(X,Y) / (sd(X) * sd(Y))

• • b) Cov(X,Y) * Var(X)

• • c) Var(X) - Var(Y)

• • d) E[X] + E[Y]

If X ~ N(µ,σ^2 ), then which of the following is true?

• • a) P(X >µ) = 0.5

• • b) P(X > µ) = 1

• • c) P(X > µ) = 0

• d) The probability of X being negative is always 0.

If two random variables are uncorrelated, then:

• • a) They are always independent.

• • b) Their covariance is zero.

• • c) Their variances must be equal.

• • d) Their sum always follows a normal distribution.