4

False; Independence is not a requirement for you to be able to find the PDF of the sum of random variables. However, independence does make it very easy to find the joint PDF of random variables, because if you know they are independent, then the joint density is just given by the product of their marginal densities.

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.

This is true. Given that Y is represented as a sum of several random variables, X1,X2,...,Xn, then the expectation of Y is given by the sum of the expectations of X1,X2,...,Xn. For example, if Y=X1+X2+X3, then E[Y]=E[X1]+E[X2]+E[X3]. This is one of the useful properties of expectation given in class.

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.

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

The "Law of Iterated Expectations" states that:

This is false. Unlike the case for expectations, the variance of a sum of random variables is equal to the sum of the variances of the random variables only when the random variables have zero covariance (which is always the case when they are independent).

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.

Infinity

In the St. Petersburg paradox discussed in class, we know the following, where Y represents winnings and X represents the number of flips required until the coin comes up heads. Winnings are defined as Y=2X,X∼G(0.5), and E(X)=2. What is the expectation of Y?

A, B; One can always find this from the CDF: If we knew the CDF function F(x), we can always calculate P(a<X<b)=F(b)−F(a). The PDF tells you either the point probabilities (for a discrete variable) or their densities (for a continuous variable), so they can always be summed or integrated to find the corresponding CDF which can then be used to calculate P(a<X<b). Neither the mean nor the median provide sufficient information. One can imagine two PDFs f(x) and g(x) with the same mean and median (e.g. two symmetric bell curves centered at 0), except that f(x) is concentrated at its mean/median (e.g. a bell curve with a small standard deviation) and g(x) is more spread out (e.g. a bell curve with a large standard deviation). These PDFs will calculate different cdfs F(x) and G(x), so For fixed a,b, will calculate different probabilities P(a<X<b).

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. Which of these will provide you enough information to find the probabilities? (Select all that apply) A. CDF B. PDF C. Mean D. Median

Measure of dispersion; same units Standard deviation is calculated as the square root of variance, and is another way of measuring the dispersion of a random variable. Standard deviation can sometimes be a convenient measure of dispersion, since it is in the same units as the random variable. Variance is also a measure of dispersion, but is in the square of the unit of the random variable (which can be seen by the fact that calculating variances involve squaring expectations of the random variable)

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

False; Since X is a binomal distribution, X is a discrete random variable. This implies that FX is not invertible, and hence you cannot use the integral transformation method, because you cannot solve for X, since the inverse is not defined.

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; As Professor Ellison explained in class, whatever the support of X,Y lives on [0,1]. This is because the Y is a CDF of a random variable..

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].

nth; The selling price in the auction would be the maximum bid from n bids. This is exactly the nth-order statistic.

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?

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, i.e. it does not hold that if the covariance between two variables is equal to zero, the two variables are independent.

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.

No, units! The problem is that the unit of covariance is the unit of the random variable, and in general we don't know that the two sets of random variables that we find pairwise covariances for have the same units. For example, for X,Y such that Cov(X,Y)=4, consider A=10X,B=10Y. Then one can see that A and B are related to each other "equally" compared to X,Y (effectively, we are just changing units: for example, X,Y could be a random variables capturing heights in centimetres, whereas A,B capture the same heights in millimetres). However, Cov(A,B)=Cov(10X,10Y)=10∗10Cov(X,Y)=400. However, correlation helps "standardize" changes of units; so for the example above, it is true that Cor(A,B)=Cor(X,Y). So correlations do let us compare the relatedness of one pair of random variables with a different pair of random variables.

Suppose random variables X,Y are such that Cov(X,Y)=4, whereas random variables A,B are such that Cov(A,B)=400. Is it correct to say that A and B are therefore more "closely related" than X and Y?

True

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?

50

Suppose that you have a function, Y=5X+2, and the variance of X is 2. What is the variance of Y?

100sig^2

Suppose the maximum score on an exam is 10. Exam scores can therefore be modeled as a random variable X. Assume this variable has variance σ2. However, in the final gradebook the exam scores will be linearly scaled so that each exam score is instead out of 100 (e.g. if one's score was originally 9, then their scaled score becomes 90). What is the variance of the random variable Y corresponding to the scaled scores?

1st; The minimum time spent running determines when the race ends, so the 1st order statistic is relevant.

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?

nore more than .5

Suppose you have a probability distribution for a non-negative random variable X, where the expectation of X is given by E[X]=5. By Markov's inequality, the probability that X is greater than or equal to 10 is _________________.

B; As per the probability integral transformation, we need to be able to sample from U[0,1], together with F−1Y, in order to do pseudorandom number generation of Y. Basically for each u∈U[0,1] that we sample from U[0,1], we calculate y:=F−1Y(u) as the corresponding sample from Y

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? (Select all that apply) A. The integral of F−1Y B. The ability to sample from standard uniform distribution U[0,1] C. The derivative of F−1Y D. The graph of the cdf of the standard uniform distribution U[0,1]

Binomial; n = 1 The Bernoulli distribution is a special case of the binomial distribution where n=1. In other words, whereas the binomial distribution might describe the outcomes of a series of coin flips, for example, the Bernoulli distribution would describe the outcome of a single coin flip.

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

1/p

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?

B; In the lecture we learned that transforming a continuous random variable by its CDF yields a random variable that is uniformly distributed. The answer choices about the uniformly distibuted functions are wrong, most easily seen from the fact that the uniform PDF is flat whereas the uniform CDF is a step function.

The example of the probability integral transformation given in class demonstrates which of the following for continuous random variables? A. If X is a uniformly-distributed random variable, then the CDF is also uniformly distributed B. The result that if you transform a random variable by its own CDF, the resulting distribution will be uniform [0,1] C. That the PDF and the CDF are equivalent functions for uniformly-distributed random variables D. The nature of the relationship between the PDF and the CDF for all types of distributions

B, D; The property Cov(aX+b,cY+d)=ac∗Cov(X,Y) implies that when you take a linear transformation of variables, the covariance is equal to the original covariance multiplied by the multiplicative constants of the linear transformation, but without consideration of the additive constants. Thus the additive constants in a linear transformation do not factor into changes in covariance. In the special case of two independent random variables, the original covariance Cov(X,Y) is 0, so any linear transformation of this necessarily leaves this unchanged.

The property that Cov(aX+b,cY+d)=ac∗Cov(X,Y) implies which of the following? (Select all that apply.) A. The covariance of a linear transformation of a set of variables is equal to the covariance of the original variables B. In a linear transformation between a set of two variables, additive constants do not factor in to any changes in covariance C. Covariance from a linear transformation of two variables is altered by both the coefficient of change and the additive constant D. Covariance is unchanged for a linear transformation of a set of two independent variables

False; As discussed in class, the expectation of winnings is infinity. You can imagine that people probably do not place an infinite value and probably not willing to pay an infinite amount to play this game. Instead, people likely have a certain utility function that describes how much they value playing the game. This utility function most likely exhibits diminishing marginal utility of money.

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; this is only true when X1 and X2 are independent

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

False

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

x

We have N i.i.d random variables from the uniform distribution between 0 and 1. If N=1, what is the probability that the nth order statistic is less than or equal to the value x? (In other words, what is Pr(X(n)1≤x)?)

B, F

What do we mean by convolution in the context of probability? (Select all that apply) A. a coil or twist, especially one of many. B. the sum of independent random variables C. any function of random variables D. all combinations and permutations of random variables E. the product of marginal PDFs F. linear combinations of independent random variables

B; The geometric distribution describes the distribution of the number of trials or attempts until a "success" is reached. For example, if you flip a coin until the coin lands heads, the number of flips that tails that would land before the first heads in repeated trials could be characterized by a geometric distribution. We will learn more about the geometric and other special distributions in the next lecture.

Which of the following describes the geometric distribution? A. The number of successes out of a given number of independent trials or attempts B. Number of identical trials repeated until a "success" is reached C. A distribution where each of the outcomes are equally likely D. A normal or "bell curve" distribution

A, D; If two variables are independent, or uncorrelated, then their covariance and correlation are both equal to zero. If two variables are negatively correlated, then correlation is less than zero and if positively correlated, then correlation is greater than zero. The other choices are incorrect. Correlation ranges between -1 to 1. Variables that are positively correlated may be weakly positive correlated, i.e. have a ρXY that is positive but closer to 0.

Which of the following statements are true? (Select all that apply) A. If two variables X and Y are independent, then the covariance, σXY, and the correlation, ρXY, are equal to zero B. Correlation ranges between 0 and 1 C. For any positively correlated variables X and Y, ρXY is close to 1 D. ρXY is greater than zero for two positively correlated variables