## Distribution in Data Science

**Discrete**

#### 1. **Binomial**

x successes in n events, each with p probability

with μ = np and σ^{2} = 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 *n ^{th }*trial

*q ^{n−}*

^{1}

*p*, with

*µ*= 1

*/p*and

*σ*

^{2 }=

^{1}

^{−p}/*p*^{2}

#### 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 *= *σ*^{2 }= *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 = Bias**^{2 }+ Variance + Irreducible Noise

^{2 }+ 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