###### Introduction to Statistics and Data Analysis: With Exercises, Solutions and Applications in R

98 SOLVED PROBLEMS

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 ...

(a) Random numbers of a normal distribution can be...
Question: 7.5

## Consider the joint PDF for the type of customer service X (0 = telephonic hotline, 1 = Email) and of satisfaction score Y (1 = unsatisfied, 2 = satisfied, 3 = very satisfied): ...

(a) The marginal distributions are obtained by the...
Question: 7.6

## Consider a continuous random variable X with expectation 15 and variance 4. Determine the smallest interval [15 − c, 15 + c] which contains at least 90 % of the values of X. ...

Using Tschebyschev’s inequality (7.24)  P\l...
Question: 7.8

## Recall the urn model we introduced in Chap. 5. Consider an urn with eight balls: four of them are white, three are black, and one is red. Now, two balls are drawn from the urn. The random variables X and Y are defined as follows: X = { 1 black ball 2 red ball in the first draw 3 white ball Y = { 1 ...

(a) The random variables X and Y are independent i...
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? ...

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” ...

P(S ≥ 1,W ≥ 1) $\overset{indep.}{=}$...
Question: 9.1

## Consider an i.i.d. sample of size n from a Po(λ) distributed random variable X.(a) Determine the maximum likelihood estimate for λ. (b) What does the log-likelihood function look like for the following realizations: x1 = 4, x2 = 3, x3 = 8, x4 = 6, x5 = 6? Plot the function using R. Hint: The curve ...

(a) The exercise tells us that X_{i} \overs...
Question: 9.2

## Consider an i.i.d. sample of size n from a N(μ, σ²) distributed random variable X. (a) Determine the maximum likelihood estimator for μ under the assumption that σ² = 1. (b) Now determine the maximum likelihood estimator for μ for an arbitrary σ². (c) What is the maximum likelihood estimate for σ²? ...

(a) The probability density function of a normal d...
Question: 11.6