Distributions

Question 1

when we use the bernoulli distribution and what is that?

Answer 1

the Bernoulli Distribution it is used in the yes-no type of problems and is express by the formula q = 1 - p.
FORMULA :
**P(X = x) = p^x (1-p)^1-x

where 0 ≤ p ≤ 1, x ∈ {0, 1}

Question 2

when we use the bernoulli distribution and what is that?

Answer 2

The binomial distribution is a discrete probability distribution that describes the number of successes (often denoted as “k”) in a fixed number of independent Bernoulli trials. Each Bernoulli trial has only two possible outcomes: success (usually denoted as “1”) or failure (usually denoted as “0”);
FORMULA
vedere la formula (..)

where
p is the probability
n= number of trials and
x is the value of the experimetnt

Question 3

A fair die is thrown four times. Calculate the probabilities of getting:
0 Twos
1 Two
2 Twos
3 Twos
4 Twos
(Exercise using binomial distribution)

Answer 3

page 16-17 of the distributions slide

Question 4

what is the discrete uniform distribution?

Answer 4

The discrete uniform distribution is a symmetric probability distribution wherein a
finite number of values are equally likely to be observed and every one of n values
has equal probability 1/n.

Question 5

what is the geometric distribution?

Answer 5

Geometric distribution is a type of discrete probability distribution that represents
the probability of the number of successive failures before a success is obtained in
a Bernoulli trial.
In other words, in a geometric distribution, a Bernoulli trial is repeated until a
success is obtained and then stopped.
FORMULA
P(X = x) = (1 − p)^x−1* p

Question 6

If your probability of success is 0.2, what is the probability you meet an
independent voter on your third try? (solve the exercise)

Answer 6

page 29 of distributions slide

Question 7

what is the poisson distribution?

Answer 7

It expresses the probability of a given number of events occurring in a fixed
interval of time or space if these events occur with a known constant mean rate
and independently of the time since the last event
FORMULA
p(x; λ) = (e^−λ * λ^x) / x!

where
λ = average number of events in a given interval
x = value of a random experiment

Question 8

what is the continuos uniform distribution?

Answer 8

the general form is
f (x) = 1/ b-a

where
x ∈ [a, b], a < b.

Question 9

what is the normal distribution?

Answer 9

The normal distribution, also known as the Gaussian distribution, is symmetrical
around its mean usually it have the form of a bell curve
FORMULA:
f (x) = (1/σ * √2π) * e^-1/2(x - µ / σ)^2

Question 10

what is the Gamma distribution?

Answer 10

see page 16 for the formula

Question 11

what is the inverse gamma ?

Answer 11

see page 17 for the formula

Question 11

what is the mean of a dataset?

Answer 11

The mean, often referred to as the average, is a measure of central tendency that represents the typical or central value of a dataset.

   n µ = ∑    xi / n
   i=1

​

​

​

​

Question 11

what is the weibull distribution?

Answer 11

f (x; α, λ) = αλ(λx) ^ α−1 exp(-(λx)^ α)

x ≥ 0, α, λ > 0

Question 11

what is the median of a dataset?

Answer 11

the median is the middle value if we sort the value.
We have two cases :
1. if the array is odd we can take the central value
2. if the array is even we have to put the two central value divided by two

Question 12

what is the mode?

Answer 12

the mode is the most frequently occured value, if there are more mode in a set, for instance two we call it bimodal

Question 12

what is the exponential distribution?

Answer 12

The exponential distribution is the probability distribution of the time between
events in a Poisson point process.
f (x; λ) = λe^−λx
where
x ≥ 0, λ > 0

Question 13

what is the expectation of a random variables

Answer 13

The expectation of a random variable, often denoted as E[x] It represents the average or mean value that you would expect from that random variable over a large number of repetitions or trials.
If we have a discrete case we use this formula

E[X]=∑xi ⋅ P(X=x)
i

Question 14

what is the variance of a random variables?

Answer 14

The variance of a random variable is a statistical measure that quantifies the spread or dispersion of the random variable’s values around its mean (expectation)

Var(X) = E[(X−μ)^2]

Question 15

what is the standard deviation of a random variables?

Answer 15

we could define it as the square root of the variance

Question 16

what is a sthocastic process?

Answer 16

A stochastic process, also known as a random process, is a mathematical concept that describes a collection of random variables evolving over time or another underlying parameter.

Question 17

what is covariance in random variables?

Answer 17

Covariance is a statistical measure that quantifies the degree to which two random variables change together, remember that is key concept also for the correlation coefficience

Cov(X,Y)=E[(X−μx)(Y−μy)]

where:
1. μx and μy are the means
2. X and Y are two random variables

Question 18

What is the correlation coefficient of a two random variables?

Answer 18

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two random variables.

r = Cov (X,Y) / σX σY

The resulting Pearson correlation coefficient:

r is a value between -1 and 1. It provides the following information:

r=1: Perfect positive linear relationship. The variables move together in a perfect linear manner.

r=−1: Perfect negative linear relationship. The variables move together in a perfect linear manner, but in opposite directions.

r=0: No linear relationship. There is no linear association between the variables.

where
σx = √Var(X) = √E[(X−μx)^2]
​σy = √Var(Y) = √E[(Y−μy)^2]

Question 19

what is the covariance matrix?

Answer 19

The covariance matrix is a square matrix that provides a comprehensive summary of the variances and covariances among multiple random variables

Remark: it is important for the PCA that we would to

Remark: in machine learning, the covariance matrix can be a valuable tool for understanding and preprocessing data, as well as for improving the performance and interpretability of models.