Question 1

Suppose a random variable $$X$$ follows the probability distribution: $$P(X=1)=0.1$$, $$P(X=4)=0.4$$, $$P(X=6)=0.4$$, $$P(X=9)=0.1. $$What is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = 5.2$$

Question 2

A fair die is rolled 18000 times. What is the approximate distribution of the number of times a '1' is rolled?

Accepted Answer

$$N(3000, 500)$$

Question 3

Suppose $$X$$ and $$Y$$ are independent random variables with $$Var(X) = 4$$ and $$Var(Y) = 9. $$What is $$Var(X+Y)$$?

Accepted Answer

$$Var(X+Y) = 13$$

Question 4

A student guesses on a 3-question true/false quiz, choosing 'true' with probability $$\frac{1}{3}$$ and 'false' with probability $$\frac{2}{3}$$ for each question independently. What is the expected number of questions answered 'false'?

Accepted Answer

$$E = 2$$

Question 5

Suppose you generate random numbers from the distribution $$P(X=1)=0.1$$, $$P(X=4)=0.4$$, $$P(X=6)=0.4$$, $$P(X=9)=0.1$$ repeatedly. Each draw takes 3 seconds. What is the expected time until you get your first '1'?

Accepted Answer

$$E = 30$$ seconds

Question 6

Given the density function $$f(x)$$ shown in the study material, what is the probability that $$U is $$between 4 and 7?

Accepted Answer

$$P(4 \leq U \leq 7) = 0.3$$

Question 7

Suppose you want to construct a 99.74% confidence interval for a sample proportion $$p$$ using the normal approximation. What is the critical $$Z$$ value you should use?

Accepted Answer

$$Z = 3.00$$

Question 8

A batch of 4 circuit boards contains exactly one defective board. An engineer and a boss each randomly select a board for testing, and both confirm their chosen boards are functional. What is the probability that a specific untested board (say, board 3) is defective?

Accepted Answer

$$P = \frac{1}{2}$$

Question 9

If $$X$$ follows a binomial distribution $$B(n,p)$$, what is the variance $$Var(X)$$?

Accepted Answer

$$Var(X) = np(1-p)$$

Question 10

Suppose you roll a fair die 5 times. You move 1 unit to the right if you get 1, and 2 units upwards otherwise. What is the probability that you are at (3,4) after 5 rolls?

Accepted Answer

$$P = \binom{5}{2} \left(\frac{1}{6}\$$right)^2$$ \left(\frac{5}{6}\$$right)^3$$

Question 11

If $$X$$ follows a geometric distribution $$Geo(p)$$, what is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = \frac{1}{p}$$

Question 12

Suppose you want to estimate the probability that the number of times you get a number less than 1 in 1000 draws from a uniform distribution between 1960 and 2000 is less than 10. Use normal approximation and the Z table. What is the approximate probability?

Accepted Answer

$$P \approx 0.0013$$

Question 13

Which of the following is a property of a density curve?

Accepted Answer

The total area under the curve is 1.

Question 14

Suppose $$X$$ and $$Y$$ are random variables with $$E[X]=2$$, $$E[Y]=3$$, and $$E[X+Y]=6. $$What is the correlation $$\rho$$ between $$X$$ and $$Y$$?

Accepted Answer

$$\rho$$ cannot be determined from the given information.

Question 15

According to the Central Limit Theorem, the sample proportion $$\hat{p}$$ approximately follows which distribution for large sample size $$n$$?

Accepted Answer

$$N\left(p, \frac{p(1-p)}{n}\right)$$

Question 16

Bayes' Theorem can be used to:

Accepted Answer

Update the probability of an event based on new evidence.

Question 17

Suppose you roll a fair die 7 times. You move 1 unit to the right if you get 1, and 1 unit to the left otherwise. What is the probability that you are at (5,0) for the first time after 7 rolls?

Accepted Answer

$$P = \binom{7}{5} \left(\frac{1}{6}\$$right)^5$$ \left(\frac{5}{6}\$$right)^2$$

Question 18

If $$X$$ and $$Y$$ are independent, which of the following is true about their variances?

Accepted Answer

$$Var(X+Y) = Var(X) + Var(Y)$$

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