Module 4: Functions and Moments of a Random Variable & Intro to Regressions

Let 𝑋 be a uniform random variable on [0,1] and let 𝑌=1𝑥. What is the CDF of 𝑦,𝐹𝑦(𝑦)?

1-(1/y)

A couple decides to continue to have children until a daughter is born. What is the expected number of children this couple will have if the probability that a daughter is born is given by 𝑝?

1/p

Suppose you and your friends are running a race and the race ends once one person has crossed the finish line. Assuming the time each person in the race spends running follows an i.i.d., which order statistic is relevant for calculating when the race ends?

1st

True or False: As is the case with expectation, the variance of a sum of random variables is always equal to the sum of the variances of the random variables.

False

True or False: For most people, the utility (or benefit) they derive from playing the St Petersburg paradox is exactly equivalent to the expected winnings from playing the game.

False

True or False: If 𝑌=𝑋1∗𝑋2, then it is always true that 𝐸[𝑌]=𝐸[𝑋1]∗𝐸[𝑋2].

False

True or False: To find the probability density function of 𝑌, one needs to integrate the expression for the CDF (obtained above) over its support.

False

Consider dependent random variables 𝑋,𝑌 defined on the same space. True or False: it is impossible to find the distribution of 𝑍=𝑋+𝑌.

False - Independence is not a requirement for you to be able to find the PDF of the sum of random variables, though it makes it easier

Suppose random variables 𝑋,𝑌 are such that 𝐶𝑜𝑣(𝑋,𝑌)=0. Then it must be the case that the variables 𝑋,𝑌 are independent.

False - It is true that if two variables are independent, then their covariance must be equal to zero. However, the relationship does not necessarily run the other way.

Suppose 𝑋 is a binomal random variable, with PMF 𝑓𝑥(𝑥) and CDF 𝐹𝑋(𝑥). Let 𝑌=𝐹𝑋(𝑋). True or False: You can use the probability integral transformation method to find out how 𝑌 is distributed.

False - Since 𝑋 is a binomal distribution, 𝑋 is a discrete random variable. This implies that 𝐹𝑋 is not invertible

Suppose 𝑋 is a continuous random variable, and is distributed uniformly over the interval [0,75] Let 𝑌=𝐹𝑋(𝑋). True or False: The induced support, or range of 𝐹𝑋 is also [0,75].

False - whatever the support of 𝑋,𝑌 lives on [0,1].

The property that 𝐶𝑜𝑣(𝑎𝑋+𝑏,𝑐𝑌+𝑑)=𝑎𝑐∗𝐶𝑜𝑣(𝑋,𝑌) implies what?

In a linear transformation between a set of two variables, additive constants do not factor in to any changes in covariance AND Covariance is unchanged for a linear transformation of a set of two independent variables

Which of the following describes the geometric distribution?

Number of identical trials repeated until a "success" is reached

Say you want to find the probabilities 𝑃(𝑎<𝑋<𝑏) for any 𝑎<𝑏 and suppose you only have one of the following pieces of information. What will provide you enough information to find the probabilities?

PDF or CDF

Which of the following correctly describes the concept of diminishing marginal utility?

Someone who has $10,000 dollars values an additional dollar less than someone who has $100

Suppose you want to do a psuedorandom generation of a variable 𝑌 that has a cdf 𝐹𝑌 and you've calculated the inverse 𝐹−1𝑌. Per the probability integral method, What else do you need for sampling from the distribution 𝑌?

The ability to sample from standard uniform distribution 𝑈[0,1]

The "Law of Iterated Expectations" states that:

The expectation of the expectation of Y given X is equal to the expectation of Y

The "Law of Total Variance" states that:

The variance of Y is equal to the variance of the expectation of Y given X added to the expectation of the variance of Y given X

Suppose that the PDF 𝑓𝑋(𝑥) of a random variable 𝑋 is an even function. Note: 𝑓𝑋(𝑥) is an even function if 𝑓𝑋(𝑥)=𝑓𝑋(−𝑥) . Is it true that the random variables 𝑋 and −𝑋 are identically distributed?

True

True or False: In linear regression, where the relationship between 𝑋 and 𝑌 is expressed as 𝑌=𝛼+𝛽𝑋+𝑈 (where 𝛽=𝜌𝑋𝑌∗𝜎𝑦/𝜎𝑋,𝛼=𝜇𝑦−𝛽∗𝜇𝑋), 𝛼+𝛽𝑋 refers to the variation in 𝑌 that is "explained" by 𝑋, and 𝑈 refers to the variation in 𝑌 that is "unexplained," where 𝐸[𝑈]=0 and 𝐶𝑜𝑣(𝑋,𝑈)=0.

True

The Bernoulli distribution is a special case of the ________ distribution where ___________.

binomial, n=1

True or False: Variance can be positive or negative, depending on the random variable.

false - must be non-negative

Standard deviation can be a useful way to capture the ___________ of a random variable _________ as the random variable itself.

measure of dispersion ; in the same units

Suppose a car is for sale at an auction where the bids are i.i.d. You want to find out the selling price of the car (which is determined by what the highest bidder offers). Which order statistic is relevant for this situation?

nth

If you transform a random variable by its own CDF...

the resulting distribution will be uniform [0,1]

What do we mean by convolution in the context of probability?

the sum of independent random variables AND linear combinations of independent random variables

True of False:𝜌𝑋𝑌 is greater than zero for two positively correlated variables

true

True or False: If two variables X and Y are independent, then the covariance, 𝜎𝑋𝑌, and the correlation, 𝜌𝑋𝑌, are equal to zero

true

True or False: The expectation of a sum of random variables is equal to the sum of the expectations of each of those random variables.

true