Q: A marketing firm surveys 5 people, each of whom will answer 'yes' or 'no' to a question. If the probability of 'yes' is 0.3, what is the expected number of 'yes' responses?

$$ \mu = n\pi = 5 \times 0.3 = 1.5 $$

Q: Given a discrete random variable $$X$$ with possible values 0, 1, 2, and probabilities 0.2, 0.5, 0.3 respectively, what is the expected value $$E(X)$$?

$$ E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1 $$

Q: If $$X is a $$discrete random variable with $$E(X) = 2$$ and $$P(X=1) = 0.3$$, $$P(X=2) = 0.4$$, $$P(X=3) = 0.3$$, what is the variance $$\$$sigma^2$$ of X$?

$$ \sigma^2 = (1-2)^2 \times 0.3 + (2-2)^2 \times 0.4 + (3-2)^2 \times 0.3 = 0.6 $$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can only assume a countable number of values, such as the number of defective items in a batch.

Question 2

Which formula correctly represents the expected value (mean) of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(X = x_i)$$?

Accepted Answer

$$ \mu = E(X) = \sum_{i=1}^N x_i P(X = x_i) $$

Question 3

A probability distribution for a discrete variable must satisfy which of the following properties?

Accepted Answer

The sum of all probabilities must equal 1.

Question 4

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 5

A company receives 4 invoices. If the probability that an invoice is late is 0.10, what is the probability that exactly one invoice is late? (Use the binomial formula.)

Accepted Answer

$$ P(X=1) = \frac{4!}{1!3!} (0.10)^1 (0.90)^3 = 0.2916 $$

Question 6

Which of the following is the formula for the variance of a binomial distribution with parameters $$n$$ and $$\pi$$?

Accepted Answer

$$ \sigma^2 = n\pi(1-\pi) $$

Question 7

A marketing firm surveys 5 people, each of whom will answer 'yes' or 'no' to a question. If the probability of 'yes' is 0.3, what is the expected number of 'yes' responses?

Accepted Answer

$$ \mu = n\pi = 5 	imes 0.3 = 1.5 $$

Question 8

Which of the following scenarios is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective bulbs in a sample of 10 bulbs.

Question 9

Given a discrete random variable $$X$$ with possible values 0, 1, 2, and probabilities 0.2, 0.5, 0.3 respectively, what is the expected value $$E(X)$$?

Accepted Answer

$$ E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1 $$

Question 10

If $$X is a $$discrete random variable with $$E(X) = 2$$ and $$P(X=1) = 0.3$$, $$P(X=2) = 0.4$$, $$P(X=3) = 0.3$$, what is the variance $$\$$sigma^2$$ of X$?

Accepted Answer

$$ \sigma^2 = (1-2)^2 	imes 0.3 + (2-2)^2 	imes 0.4 + (3-2)^2 	imes 0.3 = 0.6 $$

Question 11

A factory tests 10 light bulbs and finds that the probability a bulb is defective is 0.05. What is the probability that exactly 2 bulbs are defective? (Use the binomial formula.)

Accepted Answer

$$ P(X=2) = \frac{10!}{2!8!} (0.05)^2 (0.95)^8 $$

Question 12

Which formula gives the number of ways to choose $$x$$ objects from $$n$$ objects (combinations)?

Accepted Answer

$$ C_{n,x} = \frac{n!}{x!(n-x)!} $$

Question 13

Which of the following is a real-world example of a discrete probability distribution?

Accepted Answer

The number of cars passing through a toll booth in an hour.

Question 14

If a probability distribution for a discrete variable $$X is $$given by $$P(X=0)=0.3$$, $$P(X=1)=0.4$$, $$P(X=2)=0.2$$, $$P(X=3)=0.1$$, what is $$P(X \geq 2)$$?

Accepted Answer

$$ P(X \geq 2) = P(X=2) + P(X=3) = 0.2 + 0.1 = 0.3 $$

Question 15

A coin is tossed 5 times. What is the probability of getting exactly 3 heads? (Assume a fair coin.)

Accepted Answer

$$ P(X=3) = \frac{5!}{3!2!} (0.5)^3 (0.5)^2 = 0.3125 $$

Question 16

Which formula correctly gives the standard deviation of a binomial distribution with parameters $$n$$ and $$\pi$$?

Accepted Answer

$$ \sigma = \sqrt{n\pi(1-\pi)} $$

Discrete Probability Distributions: Binomial Distribution and Applications

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