What is the highest value of the CDF of a distribution?

• A) Depends on X
• B) 0
• C) 1
• D) 0.5

What does the CDF of a die roll represent?

• A) Most frequent value
• B) Probability up to a certain value
• C) Total possible scores
• D) Sum of all outcomes

When do we get the product of two expectations?

• A) Both functions must be linear
• B) Functions are identical
• C) Variables are dependent
• D) Variables are independent

What type of function is used for discrete random variables?

• A) probbaility mass function
• B) density function
• C) probability density function
• D) integral function

How do we define the range of X in general form of PMF?

• A) From a to b
• B) From a fixed point to infinity
• C) Between -1 and +1
• D) From 0 to 1 only

Which value of n is used in the PMF from a to b?

• A) b/a
• B) a+b
• C) b−a+1
• D) b−a

The graph of the CDF of disccrete uniform distribution is a ---------- function.

• A) Straight line
• B) Steps
• C) U shape
• D) Bridge

What type of experiment leads to a binomial distribution?

• A) Multiple possible outcomes
• B) Repeating until failure
• C) Continuous measurement
• D) One with fixed number of trials

How is independence shown using CDFs?

• A) Evaluating limits of CDFs
• B) Multiplying pdfs
• C) Checking if joint CDF equals product of marginals
• D) Comparing variances

Which real-life object is an example of uniform distribution?

• A) Loaded die
• B) Biased coin
• C) Unfair spinner
• D) Fair die

The law of total expectation is also known as:

• A) law of simple expectation
• B) law of iterated expectations
• C) law of independent events
• D) law of conditional density

What does M(t₁, t₂) represent in probability theory?

• A) Median of X₁ and X₂
• B) Moment generating function of joint distribution
• C) Mode of X₁ and X₂
• D) Mean of X₁ and X₂

What is the condition on the probability of success in a binomial setting?

• A) Randomly chosen
• B) Changes over trials
• C) Varies by person
• D) Remains constant

What does the PMF of a discrete uniform distribution assign to each value?

• A) Random probability
• B) Maximum value
• C) Equal probability
• D) Zero value

What does the product of marginals represent if variables are independent?

• A) Conditional probability
• B) Marginal pdf
• C) Expected value
• D) Joint pdf

Sum of n independent Exponential random variables (λ) results in __________

• A) Uniform random variable
• B) Normal random variable
• C) Binomial random variable
• D) Gamma random variable

A random variable X has an exponential distribution with probability distribution function is given by f(x)= 3e-3x for x>0 = 0 otherwise Find probability that X is not less than 2?

• A) e-6
• B) e-6-1
• C) e-3
• D) e-6 -3

The mean of exponential distribution is given as __________

• A) $\mu$
• B) $(\mu)^2$
• C) $\frac{1}{\mu}$
• D) $\frac{1}{\mu^2}$

Consider a random variable with exponential distribution with λ=1. Compute the probability for P (X>3)?

• A) e-2
• B) e-1
• C) e-4
• D) e-3

Let X∼U(a=3,b=9) be a continuous uniform random variable. What is the mean and variance of X?

• A) Mean = 5, Variance = 3
• B) Mean = 6, Variance = 6
• C) Mean = 6, Variance = 3
• D) Mean = 6, Variance = 9