Probability Distributions: Master the Fundamentals of Probability Modeling

Question 1

A 95% confidence interval for the mean of a uniform distribution with known range (a, b) is:

Accepted Answer

Mean +/- 1.96 * (b - a) / 4

Answer

Lower Quartile +/- 1.96 * Interquartile Range

Answer

Median +/- 1.96 * Standard Error

Answer

Percentile +/- 1.96 * Standard Deviation

Question 2

Variance is a measure of:

Accepted Answer

The average of the squared deviations from the mean

Answer

The standard deviation squared

Answer

The probability of selecting a value that deviates from the mean

Answer

The range of possible values in a distribution

Question 3

A statistical model is considered unbiased if its:

Accepted Answer

Expected value is equal to the true value of the parameter it estimates

Answer

Predictions are always correct

Answer

Variance is small

Answer

Distribution is normal

Question 4

The likelihood function of a probability distribution is used to:

Accepted Answer

Estimate the parameters of the distribution

Answer

Calculate the probability of a given outcome

Answer

Test hypotheses about the distribution

Answer

Generate random samples from the distribution

Question 5

The symbol representing the mean of a normal distribution is:

Accepted Answer

μ

Answer

σ

Answer

σ²

Answer

ρ

Question 6

Which distribution is suitable for modeling the number of arrivals in a given time interval?

Accepted Answer

Poisson

Answer

Binomial

Answer

Exponential

Answer

Uniform

Question 7

Which distribution exhibits a constant probability density function within a defined range?

Accepted Answer

Uniform

Answer

Binomial

Answer

Poisson

Answer

Exponential

Question 8

The heights of people in a city are normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What is the probability of randomly selecting a person shorter than 155 cm?

Accepted Answer

0.0228

Answer

0.0668

Answer

0.1587

Answer

0.9772

Question 9

A call center receives an average of 5 phone calls per hour, following a Poisson distribution. What is the probability of receiving at least 3 calls in any given hour?

Accepted Answer

0.727

Answer

0.140

Answer

0.332

Answer

0.549

Question 10

Which of the following is NOT a property of the binomial distribution?

Accepted Answer

Symmetrical

Answer

Discrete

Answer

Mean = np

Answer

Variance = np × (1 - p)

Question 11

The standard normal distribution has a mean of:

Accepted Answer

0

Answer

1

Answer

-1

Answer

μ

Question 12

In a Poisson process with a rate of 5 events per hour, what is the expected waiting time between arrivals?

Accepted Answer

5 minutes

Answer

12 minutes

Answer

20 minutes

Answer

60 minutes

Question 13

In the context of the binomial distribution, what symbol represents the probability of success?

Accepted Answer

p

Answer

q

Answer

λ

Answer

μ

Question 14

Which probability distribution exhibits a bell-shaped curve that is symmetrical around the mean?

Accepted Answer

Normal

Answer

Poisson

Answer

Exponential

Answer

Uniform

Question 15

In a Poisson distribution, what is the relationship between the mean and variance?

Accepted Answer

Mean = variance

Answer

Mean > variance

Answer

Mean < variance

Answer

Variance = 0

Question 16

The exponential distribution is commonly used to model which type of phenomenon?

Accepted Answer

Time intervals between events

Answer

Number of defects in products

Answer

Age distribution of a population

Answer

Height distribution of a sample

Question 17

Which fundamental property do all probability distributions share?

Accepted Answer

They sum to 1

Answer

Their maximum value is 0

Answer

Their mean is always positive

Answer

Their variance is always less than their mean

Question 18

In which real-world scenario can a Poisson distribution be applied?

Accepted Answer

Modeling the number of phone calls received hourly

Answer

Modeling the time between bus arrivals

Answer

Modeling the average weight of newborn babies

Answer

Modeling the height of a population

Question 19

Which of the following is not a parameter of a normal distribution?

Accepted Answer

Skewness

Answer

Mean

Answer

Standard deviation

Answer

Variance

Question 20

Which distribution is most подходящий for modeling the count of successes in a fixed number of independent experiments?

Accepted Answer

Binomial

Answer

Poisson

Answer

Uniform

Answer

Exponential

Question 21

Which distribution is used to model the intervals between arrivals at a bus stop?

Accepted Answer

Exponential

Answer

Binomial

Answer

Poisson

Answer

Normal

Question 22

In a normal distribution with a standard deviation of 10 and a mean of 50, what is the mean?

Accepted Answer

50

Answer

40

Answer

60

Answer

70

Question 23

The Poisson distribution is appropriate for modeling the number of events occurring in a fixed interval of time or space if these events occur randomly and independently at a constant average rate.

Accepted Answer

True

Answer

False

Question 24

The exponential distribution is a discrete probability distribution that describes the time between events in a Poisson process.

Accepted Answer

False

Answer

True

Question 25

Which of the following is NOT a real-world application of the uniform distribution?

Accepted Answer

Estimating the average temperature of a given month

Answer

Modeling the distribution of shoe sizes in a population

Answer

Simulating the outcome of a fair coin toss

Answer

Modeling the waiting time for a bus

Question 26

Given a normal distribution with mean μ = 10 and standard deviation σ = 3, what is the standard deviation of the distribution?

Accepted Answer

3

Answer

2

Answer

5

Answer

10

Question 27

Which of the following is a **significant** limitation of using probability distributions?

Accepted Answer

They rely on assumptions that may not always hold in practice.

Answer

They can only be used to model simple phenomena.

Answer

They require complex mathematical calculations.

Answer

They are not useful for making predictions.

Question 28

Which probability distribution describes the number of successes in a sequence of independent experiments?

Accepted Answer

Binomial distribution

Answer

Exponential distribution

Answer

Normal distribution

Answer

Poisson distribution

Question 29

A normally distributed random variable X has a mean of 10 and a standard deviation of 2. What is the probability that X is less than 8?

Accepted Answer

0.1587

Answer

0.5

Answer

0.8413

Answer

0.3413

Question 30

The time between arrivals of customers at a supermarket follows an exponential distribution with a mean of 5 minutes. What is the probability that a customer arrives within the next 2 minutes?

Accepted Answer

0.2642

Answer

0.7358

Answer

0.9502

Answer

0.4883

Question 31

Which of the following is a key property of the normal distribution?

Accepted Answer

Symmetrical around the mean

Answer

Skewed to the right

Answer

Its mean is greater than its median

Question 32

A survey found that the average height of a certain population is 170 cm with a standard deviation of 10 cm. Assuming a normal distribution, what percentage of the population is shorter than 160 cm?

Accepted Answer

15.87%

Answer

84.13%

Answer

34.13%

Answer

50%

Question 33

The probability of getting a tail when flipping a fair coin is:

Accepted Answer

0.5

Answer

1

Answer

0.25

Answer

0.75

Question 34

Which of the following is a key assumption of the binomial distribution?

Accepted Answer

Independent trials

Answer

Normal distribution of the population

Answer

Continuous outcome

Question 35

Given a normal distribution with a standard deviation of 5 and a mean of 10, what is the value of the mean?

Accepted Answer

10

Answer

5

Answer

25

Answer

15

Question 36

Which probability distribution is typically used to model the number of occurrences of a specific event within a fixed time interval or space?

Accepted Answer

Poisson distribution

Answer

Exponential distribution

Answer

Uniform distribution

Answer

Binomial distribution

Question 37

In an exponential distribution with a rate parameter of 1, what is the probability of observing a value less than 2?

Accepted Answer

1 - e^(-2)

Answer

e^(-2)

Answer

1 + e^(-2)

Answer

-e^(-2)

Question 38

Which probability distribution is characterized by a constant probability across a defined interval?

Accepted Answer

Uniform distribution

Answer

Poisson distribution

Answer

Binomial distribution

Answer

Normal distribution

Question 39

What is the variance of a Poisson distribution with a mean of 5?

Accepted Answer

5

Answer

10

Answer

25

Answer

50

Question 40

What is the probability of rolling a 6 on a fair six-sided die?

Accepted Answer

1/6

Answer

2/6

Answer

1/12

Answer

1/3

Question 41

Which distribution is commonly used to model the time elapsed between consecutive events in a Poisson process?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Uniform distribution

Answer

Normal distribution

Question 42

The sum of the probabilities of all possible outcomes in any probability distribution is equal to what value?

Accepted Answer

1

Answer

Greater than 1

Answer

Less than 1

Answer

0