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Which of the following is a simple representation for $\cap_{n=1}^{\infty}( a - \tfrac{1}{n} , b + \tfrac{1}{n})$, where $a < b + 1 \in \R$.
A. $(a,b)$ A. $(a,b]$ A. $[a,b)$ A. $[a,b]$
Which one of the following are alternate representations of $B \cup (A \cap C)$
A. $(B \setminus A) \cup (B \setminus C)$
A. $ (B \cup A) \cap C$
A. $ (B \cup A) + (B \cup C)$
A. $ (B \cup A) \cap (B \cup C)$.
We roll a die and if an even number shows up, we toss a coin 2 times and if an odd number shows up we toss a coin once. Which of the following is a reasonable sample space for this experiment?
A. $\W = { 1HT, 2HT, 3HT, 4HT, 5HT, 6HT }$ A. $\W = { 1H, 2HT, 3H, 4HT, 5H, 6HT }$ A. $\W = { 1H, 1T, 2HT, 2TT, 2HH, 2TH, 3H, 3T, 4HT, 4TT, \ldots$ ${\ldots,4HH, 4TH, 5H, 5T, 6HT, 6TT, 6HH, 6TH }$ A. $\W = { 1H, 1T, 2HT, 3H, 3T, 4HT, 5H, 5T, 6HT }$
Suppose you have a biased coin with probability $p$ that you toss 8 times. What is the probability that you see exactly $6$ heads?
A. $\frac{6}{8}$ A. $(p (1-p))^8$ A. $\sum_{i=1}^8 \binom{8}{i} p^i (1-p)^{8-i}$ A. $\binom{8}{6} p^6 (1-p)^{2}$
Suppose you’re in a queue, and your waiting time is given by the exponential distribution with parameter $1$. Given that you waited less than five units of time, what’s the probability that you waited less than one unit of time?
A. $1 - \int_1^{5} e^{s} ds$. A. $\int_1^{\infty} s ds \times \int_0^{5} s ds$ A. $\int_0^{1} e^{s} ds / \int_0^{5} e^{s} ds$ A. ${\int_1^{\infty} e^{s} ds/\int_5^{\infty} e^{s} ds}$
Which of the following can be pdfs of random variables?
A. $f(x) = \frac{c}{1+x^2} \quad x \geq 0$, $f(x) = 0$ otherwise. $c$ is some constant. A. $f(x) = \frac{\log(2)}{x} \quad x \geq 1,~f(x) = 0$ otherwise. A. $f(x) = \frac{c}{x^2} \quad x \geq 0,~f(x) = 0$ otherwise. $c$ is some constant. A. $f(x) = -1 \quad -1 < x < 0$, $f(x) = 2 \quad 0 < x < 1$.
Suppose $X$ is a random variable with pmf $f(k) = p (1-p)^{k-1}, \quad k = 0,1,\ldots$. Then its expectation is given by:
A. $\E[X] = p$. A. $\E[X] = \frac{1}{p^2}$. A. $\E[X] = \frac{1}{p}$. A. $\E[X] = \log(p)$.
Suppose $X$ models the number of car accidents that occur on I-15 in a month. The probability that any individual car has an accident is $0.01$. Suppose $100,000$ cars pass through the interstate on that month. Then, a good approximation for the distribution of $X$ is
A. $\Poisson(2)$ A. $N(1000,1)$. A. $\Geometric(1000)$. A. $\Poisson(1000)$.
Suppose the correlation coefficient between two random variables was determined to be $\rho(X,Y) = 0.95$. Which of the following statements are true?
A. There exist constants $a$ and $b$ such that $Y = a X + b$. A. $Y$ and $X$ are independent. A. The correlation between $X$ and $Y$ is low. A. The correlation between $X$ and $Y$ is high.
Suppose you have $100$ balls and 50 boxes. For each ball, you pick a box uniformly randomly and throw a ball in a box. Let $X$ be the number of empty boxes. Let $X_i$ be a Bernouilli random variable that is $1$ if box $i$ is empty. We know that represents the random number of empty boxes. Which of the following is $\E[X]$, the expected number of empty boxes:
A. $100 \left( \frac{49}{50} \right)^{50}$ A. $50^2 \left( \frac{99}{100} \right)^{100}$ A. $50 \left( \frac{49}{50} \right)^{100}$ A. $102.51$.
Solve on the answer book. You may get scratch paper from me, but I won’t collect it.
Suppose $X \sim N(0,1)$ is a normally distributed random variable. We’re going to find a general formula for $\E[X^n]$.
When $i = 2k$ is even, $\E[X^{2k}]$ may be computed as follows using integration by parts with $u = x^{2k-1},~dv = x e^{-x^2/2}$.
Write down a formula for $\E[X^{2k}]$ and prove it.
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Suppose $(X,Y)$ are uniformly distributed on the right triangle with vertices at $(0,0)$,$(1,0)$ and $(1,1)$.
Hint The area of a triangle is $\frac12 \text{base } \times \text{ height}$.
- What proportion of the actual population voted.
- A random person is picked. Given that she voted, what is the probability that she is an independent?
<div class="pagebreak"> </div> 1. Suppose that $X \sim \Exponential(1)$.
- Compute $M(s) = \E[e^{s X}]$
- Consider the following computation
$$
\begin{align}
\frac{d}{ds} M(s) & = \E\left[ \frac{d}{ds} e^{sX} \right] = \E[ X e^{sX} ]. \\
\Rightarrow \left. \frac{d}{ds} M(s) \right|_{s = 0} & = \E[ X ]
\end{align}
$$
Can you use this differentiation idea to find $\E[X^3]$? If you figure out how to do this correctly, you will see why $M(s)$ is called the moment generating function of $X$.
Midterm I ———-
Let $A, B$ and $C$ be events in a sample space $\W$. Express the following event in terms of $A$ $B$ and $C$ using intersections, unions, complements and set substraction.
An election in the state of Unaria has two candidates, Bjørn Sanderson and Dieter Trumpf. A sample of 3 people were polled and asked who they were voting for. Give me a sample space for the outcome of the poll. How big is $\W$?
Prove the following statement: For any number $a \in \R$,
Under what conditions on $A$ and $B$ can we have
Hint: Draw a picture.
Find the coefficient of $x^{5}y^{27}$ in $(x+3y)^{32}$.
A license plate is made up of 4 numbers (from 0 through 9) followed by 3 uppercase letters (from A-Z).
a. How many different license plates are possible? a. Let $A$ be the set of all license places that contain at least one $9$. What is $|A|$? a. You go to the DMV to pick up the first ever license plate issue by the state of Unaria. They pick a license plate uniformly randomly from all possible license plates.
Your numerologist has told you that the number $9$ is very lucky for you. As a devout practitioner of numerology, you say your prayer to the number god by calculating the probability that you'd get a license plate containing the number $9$.
<p></p> <!-- ### Independence -->
An urn contains $5$ blue balls and $15$ red ones. You pick up one ball, put it back in the urn and then pick up a second. Let
a. Are $A$ and $B$ independent events? Prove your answer.
a. Now repeat the above experiment, but this time, you hold on to the first one while picking up the second. Are $A$ and $B$ independent events? Prove your answer.
a. In the case where we do not replace the first ball, what is the probability that the first ball is blue, given that the second ball is red?
The daily weather is of Unaria dichotomous, much like its political system. It either rains all day with probability $p$ or is sunny all day with probability $1-p$.
You’re a budding Unarian meteorologist and you track the weather of Unaria in your experiment for 30 days. Assume that the weather on each day is independent of all the other days.
a. Give me a sample space for the experiment. a. Define a probability on the sample space. a. Let $X$ be the random variable that represents the number of sunny days over a 30 day period. What kind of random variable is $X$? Write down its distribution. a. What is the probability that there are $5$ sunny days in a month?
Binomial theorem
Bernoulli distribution Let $X \sim \operatorname{Bernoulli}(p)$. Then,
Binomial distribution Let $X \sim \operatorname{Binomial}(n,p)$. Then,
Geometric distribution Let $X \sim \operatorname{Geometric}(p)$. Then,