Chapter 7 Probability Distributions

Question 1

What is a random variable? P 64

Answer 1

A random variable is a quantity that is produced by a random process.

Question 2

Upper-case letters like X denote a random variable, while lower-case letters like x denote the value that the random variable takes. True/False? P 64

Answer 2

True

Question 3

What are the values of a random variable called? In which formats can they exist?P 64

Answer 3

The values that a random variable can take is called its domain, and the domain of a random variable may be discrete or continuous.

Question 4

What is a random variable that has values true or false called? P 64

Answer 4

Boolean random variable

Question 5

What are the 4 most important moments of a distribution? P 65

Answer 5

Two important properties of a probability distribution are the expected value and the variance. Mathematically, these are referred to as the first and second moments of the distribution. Other moments include the skewness (3rd moment) and the kurtosis (4th moment).

Question 6

What is the expected value? P 65

Answer 6

The expected value is the average or mean value of a random variable X. This is the most likely value or the outcome with the highest probability.

Question 7

How is expected value denoted? P 65

Answer 7

It is typically denoted as a function of the uppercase letter E with square brackets: for example, E[X] for the expected value of X or E[f(x)] where the function f() is used to sample a value from the domain of X.

Question 8

What is the Probability Distribution Function called for discrete variables? P 65

Answer 8

The probability mass function, or PMF, defines the probability distribution for a discrete random variable.

Question 9

What is Cumulative Distribution Function? P 66

Answer 9

Cumulative probability less than or equal to a value for a random variable. This is a function that assigns a probability that a random variable will have a value of less than or equal to a specific discrete value.

Question 10

How is expected value calculated? P 66

Answer 10

The expected value for a discrete random variable can be calculated from a sample using the mode, e.g. finding the most common value.

Question 11

The sum of probabilities in the PMF equals to one. True/False? P 66

Answer 11

True. Probability Mass Function. Probability for a value for a discrete random variable so sum of the probability of all the values, equals 1

Question 11

What are 3 well-known discrete probability distributions? P 66

Answer 11

Some examples of well-known discrete probability distributions include:
ˆ Bernoulli and binomial distributions.
ˆ Multinoulli and multinomial distributions.
ˆ Poisson distribution.

Question 12

What is the difference between Bernoulli and Binomial distribution? External

Answer 12

Bernoulli deals with the outcome of the single trial of the event, whereas Binomial deals with the outcome of the multiple trials of the single event. Bernoulli is used when the outcome of an event is required for only one time, whereas the Binomial is used when the outcome of an event is required multiple times.

Question 13

What is the difference between Multinoulli and Multinomial distribution? External

Answer 13

Multinoulli can be used to describe the outcome of one single random variable, which may take outcomes. On the other hand, Multinomial generalizes the multinoulli distribution, making it possible to model trials of random variable , which can take outcomes.

Question 13

What is Poisson distribution? External

Answer 13

A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space. Like the number of times a car passes a certain point on the street each hour

The Poisson distribution has only one parameter, λ (lambda), which is the mean number of events.

Question 14

What are 3 well-known continuous probability distributions? P 67

Answer 14

As a continuous function, the structure forms a smooth curve. Some examples of well-known continuous probability distributions include:
• Normal or Gaussian distribution.
• Exponential distribution
• Pareto distribution.

Question 15

What’s probability mass function (PMF)?

External

Answer 15

A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value.

Ref

Question 16

The probability mass function is defined for ____ random variables. ____ is the equivalent of the probability distribution function for the continuous random variables, it gives the likelihood of a certain random variable to assume a certain value.

External

Answer 16

discrete, Probability density function

Ref
A probability distribution may be either discrete or continuous. so a probability distribution function can be either PMF or PDF