Cumulative distribution functions

1.3.7 · AI-ML › Probability & Statistics

Cumulative Distribution Function kya hota hai?

Key properties jo har CDF mein honi chahiye:

1. Monotonically non-decreasing: Agar $x_1<x_2$ hai, toh $F_X(x_1)\le F_X(x_2)$
2. Limits: ${\lim }_{x\to -\infty }{F}_X(x)=0$ aur ${\lim }_{x\to \infty }{F}_X(x)=1$
3. Right-continuous: ${\lim }_{h\to 0^{+}}{F}_X(x+h)=F_X(x)$

CDF ko First Principles se Derive karna

Discrete Random Variables ke liye

PMF se shuru karo: Maano humara ek discrete random variable $X$ hai jiska probability mass function $p_X(x)=P(X=x)$ hai.

Step 1 — $P(X\le x)$ ka matlab kya hai?
Hum $x$ tak aur $x$ shamil karke saare outcomes ki total probability chahte hain. Kyunki disjoint events ki probabilities add hoti hain:

$F_X(x)=P(X\le x)={\sum }_{x_i\le x}{p}_X(x_i)$

Yeh step kyun? Hum individual point probabilities accumulate kar rahe hain. Har woh value $x_i$ jo $\le x$ hai, apni probability mass contribute karti hai.

Continuous Random Variables ke liye

PDF se shuru karo: Continuous $X$ ke liye jiska probability density function $f_X(x)$ hai, probabilities curve ke neeche areas hoti hain.

Step 1 — Probability as an integral:
Kyunki $P(a<X\le b)={\int }_a^b{f}_X(x)dx$ hai, $-\infty$ se $x$ tak ki cumulative probability hai:

$F_X(x)=P(X\le x)={\int }_{-\infty }^x{f}_X(t)dt$

Yeh step kyun? Hum point $x$ tak, bilkul left se, saare infinitesimal probability "slices" $f_X(t)dt$ integrate (accumulate) kar rahe hain.

Step 2 — CDF se PDF recover karna:
Fundamental Theorem of Calculus se:

$f_X(x)=\frac{d}{dx}{F}_X(x)$

Yeh relationship kyun? PDF cumulative probability ki rate of change hai. CDF antiderivative hai.

CDF se Probabilities Compute karna

Quantiles aur Percentiles se Connection

ML mein hum kyun care karte hain?

• Confidence intervals: "95% predictions is threshold se neeche hain"
• Random sampling: $X=F^{-1}(U)$ ke through samples generate karo jahan $U\sim \text{Uniform}(0,1)$
• Outlier detection: 99th percentile se aage ke data points flag karo

Properties aur Relationships

Machine Learning mein Applications

1. Model Evaluation:

   • Prediction intervals compute karna: $P(y_{\text{pred}}\in [a,b])$
   • Calibration curves predicted CDF ko empirical CDF se compare karti hain

2. Data Preprocessing:

   • Quantile normalization: CDF ke through data ko uniform distribution par map karo
   • Outlier detection: woh points flag karo jahan $F_X(x)>0.99$ ya $<0.01$ ho

3. Generative Models:

   • Inverse transform sampling: Uniform random numbers use karke kisi bhi distribution se samples generate karo
   • Copulas: Marginal CDFs combine karke joint distributions model karo

4. Statistical Tests:

   • Kolmogorov-Smirnov test empirical CDF ko theoretical CDF se compare karta hai
   • p-values tail probabilities hain: $p=1-F_{\text{null}}(t_{\text{obs}})$

Connections

• Probability Distributions — CDF, PDF ka integral hai
• Random Variables — CDF kisi bhi random variable ke liye define hota hai
• Expected Value and Variance — CDF use karke $E[X]$ compute kar sakte hain
• Inverse Transform Sampling — Random samples generate karne ke liye $F^{-1}$ use karta hai
• Quantile Functions — CDF ka inverse
• Empirical Distribution — CDF ka sample-based estimate
• Kolmogorov-Smirnov Test — Goodness-of-fit ke liye CDFs compare karta hai
• Copulas — Dependence model karne ke liye marginal CDFs join karta hai
• Survival Analysis — Survival function $S(x)=1-F_X(x)$ use karta hai

#flashcards/ai-ml

CDF $F_X(x)$ kya represent karta hai? :: Woh probability ki random variable $X$ $x$ se less than or equal to ho: $F_X(x)=P(X\le x)$

Concept Map