Distribution in Data Science


Discrete
1. Binomial
x successes in n events, each with p probability
with μ = np and σ2 = npq
= | binomial probability | |
= | number of times for a selected outcome within n trials | |
= | number of combinations | |
= | probability of success on one trial | |
= | probability of failure on a one-trial | |
= | number of trials |

Note: If n = 1, this can be a Bernoulli distribution
2. Geometric
Geometric distribution may be a variety of opportunity distribution supported three key assumptions. These are arranged as follows.

- The tests performed are independent.
- There are often only two results for every trial – success or failure.
- The probability of success, indicated by p, is that the same for every test.
first success with p probability on the nth trial
qn−1p, with µ = 1/p and σ2 =1−p/p2

3. Negative Binomial
- A negative binomial distribution (also called the Pascal Distribution) for random variables in a negative binomial experiment.
- number of failures before r successes
4. Hypergeometric
- The hypergeometric distribution is very the same as the statistical distribution. In fact, Bernoulli distribution is a superb measure of hypergeometric distribution as long as you create a sample of fifty or less of the population.


- K is that the number of successes within the population
- k is that the number of observed successes
- N is that the population size
- n is that the number of draws
- X is items of that feature
5. Poisson

number of successes x in a hard and fast quantity, where success occurs at a median rate

µ = σ2 = λ
Continuous
1. Uniform

all values between a and b are equally likely
f(x)=1/(b−a)
for a ≤ x ≤ b
Theoretical definition formulas and standard deviations are present
μ=(a+b)/2 and σ=√(b−a)2/12
2. Normal/Gaussian


= | Probability density function | |
= | Standard deviation | |
= | Mean |
Central Limit Theorem – sample mean of i.i.d. data approaches Gaussian distribution.
Empirical Rule – 68%, 95%, and 99.7% of values lie within one, two, and three standard deviations of the mean.
Normal Approximation – discrete distributions like Binomial and Poisson may be approximated using z-scores when np, nq, and λ are greater than 10
3. Exponential


= | probability density function | |
= | rate parameter | |
= | Random variable |
memoryless time between independent events occurring at a median rate λ → λe−λx, with µ = 1/λ

4. Gamma
time until n independent events occurring at a mean rate λ

where p and x are continuous chance variable.
Γ(α) = Gamma function
Concepts
Prediction Error = Bias2 + Variance + Irreducible Noise
1. Bias

wrong assumptions when training → can’t capture underlying patterns → underfit
2. Variance
sensitive to fluctuations when training→ can’t generalize on unseen data → overfit
The bias-variance tradeoff attempts to attenuate these two sources of error, through methods such as:
– Cross-validation to generalize to unseen data
– Dimension reduction and have selection
In all cases, as variance decreases, bias increases.
ML models may be divided into two types:
– Parametric – uses a hard and fast number of parameters with regard to sample size
– Non-Parametric – uses a versatile number of parameters and doesn’t make particular assumptions on the data
3. Cross-Validation
validates test error with a subset of coaching data, and selects parameters to maximize average performance-
– k-fold – divide data into k groups, and use one to validate
– leave-pout – use p samples to validate and also the rest to train
Ravi Kumar
Awesome
Mukul Gupta
Thankyou Ravi
Shreyas Baksi
Very informative
Mukul Gupta
thankyou Shreyas Baksi
Shubham Gupta
Concepts are well explained and easy to understand…!!
Mukul Gupta
Thankyou
Aman
Great👍👍
Vansh Gupta
Nice work👍
Jitendr
Good work 👍
Chirag kr vasav
nice and informative blog
Kishore kumar
Very well explained. Great work keep it up
Mritunjay Kumar Singh
Very well conceptualized. Great work sir👍
Mukul Gupta
Thankyou
Piyush Gupta
Very detailed and explained. Good job