Chapter III Part I - Statistics I - Exercises

This is a set of exercises created by the teaching Faculty for Statistics I, Chapter III, part I - discrete probability models, from the Lisbon Accounting and Business School.

Questions

Question 1

Using the Binomial tables, or the calculator, answer:

1. Verify the validity of the following claims:
   1. TRUEFALSE \(P(X\leq 3)=0.4826\) when \(X\sim Bin(9,0.4)\)
   2. TRUEFALSE \(P(2\leq X\leq 6)=0.3294\) when \(X\sim Bin(7,0.3)\)
   3. TRUEFALSE \(P(X=7)=0.2668\) if \(X\sim Bin(10,0.7)\)
2. Check the values of \(n\) and \(p\) for the following cases:
   1. TRUEFALSE \(X\sim Bin(10,0)\) and \(P(X\leq 5)=0.8338\), then \(p=0.4\)
   2. TRUEFALSE \(X\sim Bin(n,0.2)\) and \(P(2\leq X\leq n)=0.5638\), then \(n=8\).

Question 2

For a given company, the historical data indicates that the probability of a receipt being filled incorrectly is 0.2. Suppose you pick randomly 10 for analysis. The probability that:

1. Exactly 5 receipts contain irregularities is 0.10.02640.9736None
2. At most 3 receipts contain irregularities is 0.87910.12090.2013None
3. At least one was filled incorrectly is 0.10740.62420.8926None

Question 3

The National Health Service (SNS in Portugal) informs that the probability that a resident carries a gene that triggers some disease is 0.1. In Porto you randomly select 10 residents, and in Lisbon 8 to conduct a study.

1. TRUEFALSE The probability that in Lisbon 2 residents carry the gene, knowing that between both cities 5 subjects actually carried the gene, is 0.2941.

Question 4

In the supermarket SweetDrop, the probability that a customer spends more than 25 euro is 0.1.

1. If in a group of \(n\) customers, randomly chosen, it is expected that 45 spend more than 25 euro, than \(n\) is equal to 454500None
2. From a group of 20 customers, randomly chosen, you know that 10 bought products from the in-house SweetDrop brand. If we choose randomly and without replacement 5 customers, and being \(X\) a random variable that represents the number of customers that, among the chosen, buy products branded SweetDrop, then \(P(X=2)\) is 0.34830.31250.5000None

Question 5

A given sport footwear company imports a share of the inputs used in production. This material comes in boxes of 1000 units. In order to prevent counterfeit units, in each of these boxes, a random sample of 5 units is taken. If one of these raises concerns, the box is sent back to the supplier.

1. The probability that a box with 10 suspicious units is returned is 0.95090.04910.010None
2. TRUEFALSE To compute the previous probability it is important to know if the selection was made with replacement, because the difference between both values is significative.
3. The probability of having to inspect 15 units, independent among each other, until finding one that is suspect is approximately 0.0090.0000.050None
4. The expected number of units to inspect until find the first suspicious is 1050100None

Question 6

Using Poisson distribution tables, or a calculator:

1. Check the value of the following probabilities:
   1. TRUEFALSE \(P(X\leq 5)=0.0671\) if \(X\sim Poi(10)\)
   2. TRUEFALSE \(P(4\leq X\leq 8)=0.4914\) if \(X\sim Poi(5)\)
   3. TRUEFALSE \(P(X=7)=0.0901\) if \(X\sim Poi(10)\)
2. Verify the values for \(\lambda\)
   4. TRUEFALSE \(\lambda=8.1\) if \(P(X=8)=0.1321\)
   5. TRUEFALSE \(\lambda=4.3\) if \(P(X\leq 4)=0.7442\)
   6. TRUEFALSE \(\lambda=6.5\) if \(P(X\geq 6)=0.6310\)

Question 7

The number of patients arriving daily to the ICU in a hospital, follows a Poisson process with mean 4. The ICU has capacity of 6 patients, the others, are derived to the nearest hospital. Assess the validity of the following sentences:

1. TRUEFALSE The probability that, on a given day, there is no need to transfer any patient is 0.8893.
2. TRUEFALSE The most likely number of patients arriving daily to the ICU is 6.
3. TRUEFALSE The probability that, on a given day, arrive 5 patients, given that in the previous day only 2 patients arrived, is 0.1563.
4. TRUEFALSE The probability that, in 5 days, at least 15 patients arrive to the ICU is 0.8435.
5. TRUEFALSE To ensure that approx. 97% of the time (days) there are no transfers, it is necessary to increase the capacity in 4 more beds.

Question 8

Suppose that in Branch 17 of the Bank of Coins, the number of clients inquiring about financial services, is a random variable \(X\) that follows a Poisson process. You know that the probability that in the first hour no client shows up is 0.3679, and that this branch opens from 8:30 in the morning until 14:30 without any breaks. Further, you know from the data that 25% of the inquiries are made by female clients.

1. The expected value for the number of clients showing up between 12:00 and 13:00 is 0.430.11None
2. TRUEFALSE Letting \(\lambda=1\), in an hour, the probability that in a day at most 4 clients requested information, given that until then only 2 had done so is 0.9963.
3. Choosing randomly 10 clients, in some day, the probability that 6 or less of them were female is 0.99650.00350.9803None
4. TRUEFALSE The probability of having to receive 4 clients until one of the is female is 0.4219.

Question 9

You know that the probability that a subject is allergic to the active ingredient of drug M is 0.05. In the laboratory 80 randomly chosen subjects were tested to arrive to this conclusion.

1. TRUEFALSE The probability that exactly 2 subjects are allergic to the active ingredient of drug M is approximately 0.1465.

Question 10

Let \(X\) and \(Y\) be random variables, with the same expected value, 3, and distributed as:

\[X\sim Bin(10,0)\quad and \quad Y\sim Poi(\lambda)\]

1. TRUEFALSE \(E[X-2Y]=0\)
2. TRUEFALSE \(P(X+Y=5 | X<2)=0.1553\)

Question 11

1. TRUEFALSE If \(X\sim Bin(n,p)\) and \(Y\sim Bin(n,\theta)\) are independent random variables, then \(X+Y\sim Bin(n,p+\theta)\).
2. TRUEFALSE If \(X\) and \(Y\) are independent r.v.s such that \(X\sim Poi(2)\) and \(Y\sim Poi(3)\), then \(X+Y\sim Poi(5)\).