Question 8.3: A dodecahedron is a die with 12 sides. Suppose the numbers o......

Introduction to Statistics and Data Analysis With Exercises, Solutions and Applications in R [1403016]

A dodecahedron is a die with 12 sides. Suppose the numbers on the die are 1–12. Consider the random variable X which describes which number is shown after rolling the die once. What is the distribution of X? Determine E(X) and Var(X).

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

The random variable X follows a discrete uniform distribution because p $_{i}=\frac{1}{12}$ for each x $_{i}$ . The expectation and variance are therefore

E(X) = $\frac{k+1}{2}=\frac{12+1}{2}=6.5,$

Var(X) = $\frac{1}{12} \left(12^{2}-1\right)\approx 11.92.$

Related Answered Questions

Question: 8.11

Read Appendix C.3 to learn about the Theorem of Large Numbers and the Central Limit Theorem. (a) Draw 1000 realizations from a standard normal distribution using R and calculate the arithmetic mean. Repeat this process 1000 times. Evaluate the distribution of the arithmetic mean by drawing a kernel ...

Verified Answer:

(a) Random numbers of a normal distribution can be...

Question: 8.2

A study on breeding birds collects information such as the length of their eggs (in mm). Assume that the length is normally distributed with μ = 42.1 mm and σ² = 20.8². What is the probability of (a) finding an egg with a length greater than 50 mm? (b) finding an egg between 30 and 40 mm in length? ...

Verified Answer:

Given X ∼ N(42.1, 20.8²), we get: (a) P(X ≥ 50) = ...

Question: 8.10

A reinsurance company works on a premium policy for natural disasters. Based on experience, it is known that W = “number of natural disasters from October to March” (winter) is Poisson distributed with λW = 4. Similarly, the random variable S = “number of natural disasters from April to September” ...

Verified Answer:

P(S ≥ 1,W ≥ 1)

\overset{indep.}{=}

...

Question: 8.7

A country has a ratio between male and female births of 1.05 which means that 51.22 % of babies born are male. (a) What is the probability for a mother that the first girl is born during the first three births? (b) What is the probability of getting 2 girls among 4 babies? ...

Verified Answer:

The probability of getting a girl is p = 1 − 0.512...

Question: 8.4

Felix states that he is able to distinguish a freshly ground coffee blend from an ordinary supermarket coffee. One of his friends asks him to taste 10 cups of coffee and tell him which coffee he has tasted. Suppose that Felix has actually no clue about coffee and simply guesses the brand. What is ...

Verified Answer:

Each guess is a Bernoulli experiment where the rig...

Question: 8.8

A fishermen catches, on average, three fish in an hour. Let Y be a random variable denoting the number of fish caught in one hour and let X be the time interval between catching two fishes.We assume that X follows an exponential distribution. (a) What is the distribution of Y ? (b) Determine E(Y ) ...

Verified Answer:

(a) The random variable Y follows a Poisson distri...

Question: 8.9

A restaurant sells three different types of dessert: chocolate, brownies, yogurt with seasonal fruits, and lemon tart. Years of experience have shown that the probabilities with which the desserts are chosen are 0.2, 0.3, and 0.5, respectively.(a) What is the probability that out of 5 guests, 2 ...

Verified Answer:

The random variable X = “choice of dessert” follow...

Question: 8.6

Marco’s company organizes a raffle at an end-of-year function. There are 4000 raffle tickets to be sold, of which 500 win a prize. The price of each ticket is €1.50. The value of the prizes, which are mostly electrical appliances produced by the company, varies between €80 and €250, with an average ...

Verified Answer:

(a) Let X be a random variable describing “the num...

Question: 8.5

An advertising board is illuminated by several hundred bulbs. Some of the bulbs are fused or smashed regularly. If there are more than 5 fused bulbs on a day, the owner of the board replaces them, otherwise not. Consider the following data collected over a month which captures the number of days ...

Verified Answer:

(a) It seems appropriate to model the number of fu...

Question: 8.1

A company producing cereals offers a toy in every sixth cereal package in celebration of their 50th anniversary. A father immediately buys 20 packages. (a) What is the probability of finding 4 toys in the 20 packages? (b) What is the probability of finding no toy at all? (c) The packages contain ...

Verified Answer:

The random variable X: “number of packages with a ...