Probability Distributions

Question 1

What is a probability distribution?

Answer 1

It describes how probabilities are distributed over the values of a random variable.

Question 2

What properties define a normal distribution?

Answer 2

Symmetrical, bell-shaped curve, mean=median=mode, and fully described by mean and standard deviation.

Question 3

How do you calculate probabilities in a binomial distribution?

Answer 3

Using the formula involving the number of trials, success probability per trial, and desired number of successes.

Question 4

What is a Poisson distribution and where is it commonly used?

Answer 4

It models the number of events in a fixed interval of time or space and is used in fields like telecommunications and insurance.

Question 5

Describe the characteristics of a uniform distribution.

Answer 5

All outcomes are equally likely over the interval.

Question 6

How does the exponential distribution differ from the normal distribution?

Answer 6

It is used for modeling time between events and has a rapid decrease in probability from a starting point.

Question 7

What are the key features of a geometric distribution?

Answer 7

Models the number of trials until the first success and is memoryless like the exponential distribution.

Question 8

How is the standard deviation calculated in a normal distribution?

Answer 8

It is the square root of the variance, which is a parameter of the distribution.

Question 9

What does the shape parameter signify in a Weibull distribution?

Answer 9

Indicates the concentration of failure times or the shape of the distribution’s tail.

Question 10

Why is the normal distribution important in statistics?

Answer 10

Because many statistical methods assume data follow a normal distribution due to the central limit theorem.

Question 11

How can you use the central limit theorem with non-normal distributions?

Answer 11

By using it to approximate distributions of sample means from any distribution as normal when sample sizes are large.

Question 12

What is the difference between discrete and continuous distributions?

Answer 12

Discrete distributions count occurrences, while continuous distributions measure outcomes.

Question 13

Explain the concept of the cumulative distribution function (CDF).

Answer 13

It describes the probability that a random variable is less than or equal to a certain value.

Question 14

What is the significance of the mean in a probability distribution?

Answer 14

It’s the expected value or the balance point of the distribution.

Question 15

How do you find the median in a probability distribution?

Answer 15

By finding the value at which half of the observations lie above and half below.

Question 16

What role does variance play in probability distributions?

Answer 16

It measures how spread out the values are around the mean.

Question 17

What is a hypergeometric distribution?

Answer 17

It models the number of successes in a sample without replacement from a finite population.

Question 18

How do z-scores relate to normal distributions?

Answer 18

They measure the number of standard deviations an element is from the mean in a normal distribution.

Question 19

What is a negative binomial distribution?

Answer 19

It counts the number of failures before a specified number of successes occurs.

Question 20

Why might one use a log-normal distribution?

Answer 20

To model variables that are positively skewed, such as income or city sizes.

Question 21

How do you determine if a distribution is skewed?

Answer 21

By looking at the skewness coefficient or comparing the mean and median.

Question 22

What is the difference between probability and density in distributions?

Answer 22

Probability applies to discrete cases; density applies to continuous cases.

Question 23

How are percentiles calculated in a distribution?

Answer 23

By finding the values below which a certain percentage of the data fall.

Question 24

What is a hazard function in survival analysis?

Answer 24

It describes the rate at which subjects fail or die over time.

Question 25

Why are probability distributions essential in machine learning?

Answer 25

They model the underlying distributions of data features necessary for algorithms like classification and regression.

Question 26

How do you estimate parameters of a distribution?

Answer 26

Through maximum likelihood estimation, method of moments, or Bayesian estimation.

Question 27

What is a beta distribution, and when is it used?

Answer 27

Used to model variables that are constrained to intervals like percentages or proportions.

Question 28

Describe the process of hypothesis testing using a normal distribution.

Answer 28

Set up a null hypothesis about the mean, calculate the z-score for the sample mean, and compare it to a critical value.

Question 29

How do parameter estimates affect the shape of a distribution?

Answer 29

Changes in estimates can shift or stretch/compress the distribution curve.

Question 30

What is the law of large numbers, and how does it relate to distributions?

Answer 30

It states that averages of samples converge to the population mean as sample sizes increase.

Question 31

What is the difference between parametric and non-parametric distributions?

Answer 31

Parametric involves specific distribution with set parameters; non-parametric does not assume a specific distribution.

Question 32

How can you model dependencies between variables using distributions?

Answer 32

Through copulas or joint distributions that express how variables influence each other.

Question 33

What is a mixture model in statistics?

Answer 33

It represents a combination of two or more probability distributions.

Question 34

How do outliers impact the assumptions of normal distribution?

Answer 34

They can cause violations of the assumption that data are normally distributed.

Question 35

How can transformations help in normalizing distributions?

Answer 35

By using log, square root, or other transformations to make data more symmetric.

Question 36

What is the importance of tail behavior in risk assessment?

Answer 36

Tail behavior helps assess the risk of extreme outcomes in distributions.

Question 37

How do you handle overdispersion in count data?

Answer 37

By using models like negative binomial when Poisson assumptions do not hold.

Question 38

Why might someone use Monte Carlo simulations in studying distributions?

Answer 38

To model and understand the behavior of random variables and uncertainties in complex systems.

Question 39

What are the implications of heavy-tailed distributions in finance?

Answer 39

They can indicate larger risks of extreme outcomes, important for risk management.

Question 40

How do you interpret a probability plot?

Answer 40

By comparing the data’s distribution with a theoretical distribution to assess normality or other characteristics.