Q: Let $$X, Y be $$two i.i.d. random variables with $$E(X)=E(Y)=1$$, $$\text{sd}(X)=\text{sd}(Y)=1. $$What is the distribution of $$X+Y$$?

We can only conclude that $$E(X+Y)=2$$ and $$\text{Var}(X+Y)=2$$, but we cannot make conclusion on the shape of the distribution.

Question 1

You pick with replacement candies from a bag of M&Ms. You pick 50 candies and the probability of picking a red one is $$\frac{1}{5}. $$What is the distribution of the number of red M&Ms you get?

Accepted Answer

Binomial($$n=50$$, $$p=\frac{1}{5})$$

Question 2

You pick with replacement candies from a bag of M&Ms. The probability of picking a red one is $$\frac{1}{5}. $$What is the distribution of the number of M&Ms you need to pick to get a red one?

Accepted Answer

Geometric($$p=\frac{1}{5})$$

Question 3

The rate at which you get distracted when working in front of a computer connected to Internet is 1 every 40 seconds. What is the distribution of the number of times you get distracted in 1 minute?

Accepted Answer

Poisson($$\lambda=1.5)$$

Question 4

The rate at which you get distracted when working in front of a computer connected to Internet is 1 every 40 seconds. What is the distribution of the amount of time, in minutes, until the next time you get distracted?

Accepted Answer

Exponential($$0.5)$$

Question 5

Assume the lifetime of a specific brand of lightbulbs, in years, is exponentially distributed with $$\lambda=0.2. If a $$store sells 5 lightbulbs to a customer, what is the distribution of the number of lightbulbs that will stop working within 2 years of purchase among the ones purchased by the customer?

Accepted Answer

Binomial($$n=5$$, $$p=1-e$$^{-0.4}$$)

Question 6

Let $$X \sim N(0,1)$$ and $$Y \sim N(4,3) be $$two independent random variables. What is the distribution of $$X-Y$$?

Accepted Answer

$$X-Y \sim N(-4, \sqrt{10})$$

Question 7

Let $$X, Y be $$two i.i.d. random variables with $$E(X)=E(Y)=1$$, $$	ext{sd}(X)=	ext{sd}(Y)=1. $$What is the distribution of $$X+Y$$?

Accepted Answer

We can only conclude that $$E(X+Y)=2$$ and $$	ext{Var}(X+Y)=2$$, but we cannot make conclusion on the shape of the distribution.

Question 8

Let $$X \sim N(0,1)$$, $$Y \sim N(1,2)$$, $$Z \sim N(1,1) be $$three independent random variables. What is the distribution of $$X+Y+Z$$?

Accepted Answer

$$X+Y+Z \sim N(2, \sqrt{1+4+1})$$

Question 9

You go grocery shopping 40 times and record the total amount of money spent. Assume independence between trips. Which random variable will have an approximately normal distribution?

Accepted Answer

The average amount spent per trip

Question 10

Your friend thinks that 30% of students will get an A in this class, 50% will get a B, 15% will get a C, 3% will get a D, and 2% will get an F. You want to know if he is right. Which test should you use?

Accepted Answer

Chi-square test for goodness-of-fit

Question 11

Assume that in 2019 the percentage of UCSD students who spent Thanksgiving with their family was 65%. You are curious to know if the percentage decreased in 2020 because of the pandemic. What are the correct hypotheses?

Accepted Answer

$$H_0$$: p = 0.65$$, H_A$$: p \u003c 0.65$$

Question 12

You would like to know if in the United States, the amount of money spent by individuals during Black Friday differs on average between 2020 and 2021. You interview 300 random Americans and record the amount they spent in both years. Which test should you use?

Accepted Answer

Two-sample t-test

Question 13

You construct a 95% confidence interval for the average amount of calories consumed by Americans at Thanksgiving and you get [3000, 5000]. What is the correct interpretation?

Accepted Answer

If you were to repeat the process many times of getting a sample and constructing a CI, 95% of the random intervals will contain the true average amount of calories Americans consume at Thanksgiving.

Question 14

Suppose you construct a 95% CI for the average amount of calories consumed by Americans at Thanksgiving and get [3000, 5000]. You want to test $$H_0$$: \mu = 4000$$ vs H_A$$: \mu \u003c 4000$$ at $$\alpha = 0.05$$. What is the correct statement?

Accepted Answer

We fail to reject the null since 4000 is in the 95% CI.

Question 15

An article claims that on average, Americans consume 4500 calories on Thanksgiving. You conduct a hypothesis test with a sample of 40 Americans and significance level 0.05. Suppose the power of this test is 90. You repeat the test with a sample of 100 Americans. What is true about the power of the new test?

Accepted Answer

power $$\u003e 90$$

Question 16

An article claims that on average, Americans consume 4500 calories on Thanksgiving. You conduct a hypothesis test with a sample of 40 Americans and significance level 0.05. Suppose the power of this test is 90. You repeat the test with a sample of 100 Americans. What is true about the probability of type I error of the new test?

Accepted Answer

The probability of type I error stays the same

Question 17

When you go to an Italian restaurant in the United States you could order pizza, pasta, or some meat/fish dish. You are curious to know if there is a difference between what people usually order between Americans and Italians. Which test should you use?

Accepted Answer

Chi-square test for independence

Question 18

You are curious to compare the average amount of time students at UCSD dedicate to study per day between undergraduate students and graduate students. You interview 40 random undergraduate students and 20 random graduate students. Which test should you use?

Accepted Answer

Two-sample t-test

Review of Probability Distributions, Hypothesis Testing, and Statistical Inference

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