chapter5

Question 1

What is a random variable?

Answer 1

A random variable is a variable that takes numerical values based on the outcomes of a random experiment.

Question 2

What are the two types of random variables?

Answer 2

Discrete Random Variable: Takes a countable number of values (e.g., number of steps taken).
Continuous Random Variable: Takes any value within a range (e.g., time spent).

Question 3

What is the probability distribution of a discrete random variable?

Answer 3

A table or graph showing the probabilities of all possible outcomes.

Rule 1: P(x)≥0
Rule 2: ΣP(x)=1

Question 4

How is the expected value (mean) of a discrete random variable calculated?

Answer 4

The expected value is the weighted average of all possible values:
μ=E(x)=ΣxP(x)

Question 5

What is a binomial experiment?

Answer 5

A binomial experiment has:

Fixed number of trials (n)
Two outcomes (success/failure)
Constant probability of success (p)
Independent trials.

Question 6

What is the formula for a binomial probability?

Answer 6

P(x)=(n choose x)p^x(1−p)^(n−x)

Where:
n: Number of trials
x: Number of successes
p: Probability of success

Question 7

What is a Poisson distribution?

Answer 7

A distribution for the number of events occurring in a fixed interval of time or space when:

Events occur independently.
The average rate (λ) is constant.

Question 8

What is the formula for a Poisson probability?

Answer 8

P(x)=e^−λ(λ^x)/x!

Where:
λ: Mean number of events
x: Number of events
e: Base of natural logarithms (≈2.718)

Question 9

How are binomial and Poisson distributions different?

Answer 9

Binomial: Fixed number of trials, two outcomes, probability remains constant.
Poisson: Models rare events over continuous time/space, no fixed trials.

Question 10

How do you calculate variance and standard deviation for a discrete random variable?

Answer 10

Variance: σ²=Σ(x−μ)²P(x)
Standard Deviation: σ=√σ²

Question 11

What are some examples of discrete random variables?

Answer 11

Number of patients arriving at a clinic.
Number of tails in a coin toss.
Number of defective items in a batch.

Question 12

What keywords in questions help identify a binomial distribution?

Answer 12

“Fixed number of trials”
“Two outcomes: success or failure”
“Probability remains constant”
“Independent trials”

Question 13

What keywords in questions help identify a Poisson distribution?

Answer 13

“Events per unit of time/space”
“Rare events”
“Independent occurrences”
“Rate of occurrence is constant”

Question 14

What is the Empirical Rule for probability distributions?

Answer 14

For a mean (μ) and standard deviation (σ):

P(μ−σ≤x≤μ+σ)≈68%
P(μ−2σ≤x≤μ+2σ)≈95%
P(μ−3σ≤x≤μ+3σ)≈99.7%

Question 15

What is the cumulative distribution function (CDF) for a discrete random variable?

Answer 15

The CDF gives the probability that the random variable X is less than or equal to a specific value x:
F(x)=P(X≤x)=ΣP(X=k) for k≤x

Question 16

What are some real-life applications of the binomial distribution?

Answer 16

Flipping a coin a fixed number of times.
Testing a batch of products for defects.
Conducting surveys with yes/no responses.

Question 17

What are some real-life applications of the Poisson distribution?

Answer 17

Number of calls received by a call center per hour.
Number of accidents at a traffic intersection in a day.
Number of printing errors in a book.

Question 18

What is the variance of a binomial distribution?

Answer 18

The variance of a binomial distribution is:
σ²=n⋅p⋅(1−p)

Question 19

What is the variance of a Poisson distribution?

Answer 19

The variance of a Poisson distribution equals its mean:
σ²=λ

Question 20

How do you recognize if a problem involves a discrete random variable?

Answer 20

The variable involves countable outcomes (e.g., 0, 1, 2, …).
The probability of each outcome can be listed.
Examples: Number of students in a class, dice rolls.

Question 21

What is the shape of a binomial probability distribution?

Answer 21

Symmetric: When p=0.5 and n is large.
Skewed: When p is closer to 0 or 1.

Question 22

How is the Poisson distribution derived from the binomial distribution?

Answer 22

The Poisson distribution is a limiting case of the binomial distribution when:

n→∞ (large number of trials).
p→0 (small probability of success).
λ=n⋅p (mean remains constant).

Question 23

What is the relationship between mean and variance in the Poisson distribution?

Answer 23

In a Poisson distribution, the mean and variance are equal:
μ=σ²=λ

Question 24

What are the key characteristics of a probability mass function (PMF)?

Answer 24

Assigns probabilities to each value of a discrete random variable.
The sum of all probabilities equals 1.
Probabilities are non-negative.

Question 25

What is the difference between PMF and PDF?

Answer 25

PMF (Probability Mass Function): Used for discrete random variables, assigns probability to exact values. PDF (Probability Density Function): Used for continuous random variables, represents probabilities over intervals.

Question 26

How does the law of large numbers apply to random variables?

Answer 26

As the number of trials increases, the sample mean of a random variable approaches its expected value.

Question 27

What is the central limit theorem?

Answer 27

For large sample sizes, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's original distribution.

Question 28

How do you find the mode of a discrete random variable?

Answer 28

The mode is the value of X that has the highest probability P(X=x).

Question 29

What is the standard deviation of a binomial distribution?

Answer 29

The standard deviation of a binomial distribution is: σ=√(n⋅p⋅(1−p))

Question 30

How can you identify if a problem involves Poisson distribution?

Answer 30

Look for: Number of events in a fixed interval. Rare or infrequent events. No upper limit on the number of occurrences.