A student conduct a survey form which has these question:

1. 1. How much do you use the Air Conditioner in Hall in 1 day? (in minutes)

2. 2. On a scale of 1 to 10, how much do you like using Hall Aircon app?

3. 3. Do you see the excess use of Air Conditioner might have an effect on the Environment?

4. 4. Do you have any other remarks you want to share?

How many questions above would have their answer as CONTINUOUS random variable?

(a)  

A magician has a trick machine. This machine will draw a fair dice 90% of the time. For the rest 10%, a weighted dice is drawn which has a 6 for all six faces. Let X be the random variable denoting the value of the dice when drawn. What is the variance of X?

(a)  

A coin which has either value 0 or 1, and a fair dice, is rolled together. Find the variance of the sum of their outcome, if the outcome of the dice and the coin is statistically independent?

(a)  

Given a probability distribution $$p\left(X\ =\ x\right)\ =\ \frac{k}{nx}$$ and $$p\left(Y=y\right)\ =\ \frac{k+n}{y}$$ for 0 < x, y < 4. Find k-n?

(a)  

How many statements below are correct?

1. 1. Variance of the Binomial distribution equals to its mean

2. 2. The Geometric distribution calculates the probability of an event failing n-1 times and successful 1 time

3. 3. Mean of the Poisson distribution is the average number of success in a given amount of time

4. 4. Variance of the Binomial distribution will approximate to its mean when n is large and p is small

5. 5. If we do something with a success rate of 0.05, we will definitely success after 20 tries

How many of the following statements are correct?

1. 1. We can use Poisson distribution when we need to calculate what is the chance of 10 People catching a rare disease out of a city

2. 2. We can use Binomial distribution to calculate how many calls we will get during a month given the average number of calls during a day

3. 3. We can use Geometric distribution to calculate how many tries we would need to do in order to succeed, given the probability

4. 4. We can use Poisson distribution to know how many people to allocate for a service

2 soccer teams play a match with each other. For Team A, every minute, they score, in average, 1/200 goals. For Team B, every minute, they score, in average, 1/100 goals. The typical soccer match goes for 90 minutes. What is the probability that the match ends in 2-1 for Team B?

There is a mystery basket. The probability of drawing a lucky 1000$ red ball is 0.1. For each round, you pick n balls (with replacement). After 100 rounds, you manage to pick exactly 2 red balls out of n for 15 rounds. How many balls were you picking each round?

(a)  

How many of these statements are right about CONTINUOUS variables?

1. 1. Probability density function (PDF) can take any value

2. 2. Pdf can take any positive value

3. 3. Pdf can be > 1

4. 4. If Pdf > 1, that means Cdf > 1

5. 5. Because Pdf can take value >1, they can also take value <0 so that Cdf always =1

Let pdf of a continuous variable x: $$f\left(x\right)\ =\ \frac{1}{k^x}$$ for x > 0, and 0 otherwise. Find k.

(a)  

Let pdf of a continuous variable: $$f\left(x\right)\ =\ \frac{1}{k^x}$$ for x > 0, and 0 otherwise. Find the expected value of x.

(a)  

Given X, a continuous variable, following the pdf: f(x) in the picture:

k is a constant such that f(x) is a pdf in appropriate domain of x.

Calculate Pr(X = 10)

(a)  

Given the interarrival time of passenger follows the geometric function. The median arrival time is 10 minutes. Find, on average, how many people arrive in 1 hour?

(a)  

Quiz 1 has the average of 65, standard deviation of 10, and follow a bell curve (Normal distribution). If you want to get an A, you need to be above the third quartile. How many points do you need to get an A?

(a)