What is the MGF of a random variable defined as?

• A) Derivative of CDF
• B) Expected value of a constant
• C) Integral of PDF
• D) Expected value of e tx

Why is the law of total expectation important?

• A) Shows that expectations are independent
• B) Removes need for joint distributions
• C) Simplifies computations involving conditional distributions
• D) Proves variables are uncorrelated

If $X_1$ and $X_2$ are independent, the joint pdf is:

• A) zero
• B) always equal to 1
• C) equal to the product of their marginal pdfs
• D) equal to their sum

Which type of function can be used if its expectation exists?

• A) Any valid function
• B) squared functions
• C) Straight-line functions
• D) Log functions

Under what condition does the joint cumulative distribution function equal the product of the marginal CDFs?

• A) If distributions are equal
• B) When random variables are independent
• C) If variables are uncorrelated
• D) When means match

What does the expectation of a function of one variable show?

• A) Probability
• B) Limit
• C) Average value
• D) Variation

Let X and Y be two independent discrete random variables where: P(X=0)=0.5,P(X=1)=0.5 P(Y=0)=0.6,P(Y=1)=0.4 Let Z=X+Y. What is P(Z=1))?

• A) 0.5
• B) 0.4
• C) 0.46
• D) 0.3

The total area under the curve of a probability density function is:

• A) 0
• B) 1
• C) 0.5
• D) 2

Which function is used to derive the MGF of a discrete distribution?

• A) Probability density function
• B) Mean minus standard deviation
• C) Expected value of etx
• D) Variance formula

What type of series is formed in the MGF derivation of a discrete uniform distribution?

• A) Geometric
• B) Harmonic
• C) Arithmetic
• D) Fibonacci

What type of distribution is discrete uniform?

• A) Positively skewed
• B) Negatively skewed
• C) Synmmetrical
• D) Skewed

What should always be mentioned along with a pdf expression?

• A) Graph shape
• B) Integration technique
• C) Domain of the variable
• D) Mean value

Where does the CDF always start from?

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

According to the property, correlation coefficients are zero if the variables X and Y are

• A) Dependent
• B) independent
• C)
• D)

What does MGF stand for in probability theory?

• A) Mean Generating Formula
• B) Moment Growth Formula
• C) Mathematical Generating Function
• D) Moment Generating Function

What shows X₁ and X₂ are independent using MGF?

• A) Difference of marginal MGFs equals joint MGF
• B) They are always equal
• C) Product of marginal MGFs equals joint MGF
• D) Their sum equals the joint MGF

What does the CDF of a joint distribution describe?

• A) Derivative of a pdf
• B) Relationship between two variables
• C) Spread of a single variable
• D) Sum of marginal values

How many outcomes are possible in each binomial trial?

• A) At least three
• B) One or more
• C) Several categories
• D) Only two

What is the key feature of a discrete uniform distribution?

• A) All values have the same probability
• B) Probability changes with value
• C) Only one value is likely
• D) Each outcome is unique

How do you check independence using a joint pdf?

• A) Compare joint and sum of marginals
• B) Multiply the marginals and compare with the joint
• C) Integrate the marginals
• D) Differentiate joint pdf