4.9.8 · D1Common continuous distributions — Uniform, Normal, Exponenti
Ek continuous distribution ek tarika hai jisme exactly ek unit ki "probability mass" ko numbers ki ek range pe phailaaya jaata hai — jaise ek kilogram butter ko bread ke ek slice pe phailaao — jahan values likely hain wahan mota, jahan rare hain wahan patla. Parent topic mein jo kuch bhi hai woh bas yahi hai: smear ki shape chuno, phir uske neeche areas measure karo.
Parent note mein ek bhi formula padhne se pehle, tumhe us alphabet ka maalik banana hoga jisme woh bolta hai. Neeche har symbol aur idea hai jo woh use karta hai, is tarah order kiya gaya hai ki har ek sirf upar waale par depend kare. Koi cheez tab tak nahi aati jab tak kamaai na ho.
Definition Number line aur[ a , b ] number line bas sabhi real numbers ka ruler hai jo left (negative) aur right (positive) ki taraf stretch karta hai. Ek interval [ a , b ] a b
a left edge (sabse chhoti allowed value).b right edge (sabse badi allowed value).Square bracket [ ] ( )
Ise picture karo: ek horizontal ruler jisme a b
Topic ko yeh kyun chahiye: har distribution kisi na kisi line ke stretch par rehti hai — Uniform [ a , b ] [ 0 , ∞ ) [ 0 , 1 ] stage hai jis par probability phailaai jaati hai.
Definition Random variableEk random variable X X x
Ise picture karo: ek pointer jo number line par kahin drop hoga, par tumhe abhi pata nahi kahan. Experiment ke alag-alag runs mein yeh alag-alag jagah drop hota hai.
Topic ko yeh kyun chahiye: poora subject is baare mein hai ki pointer kahan land karna chahta hai . X X
P ( kuch ) 0 1 0 1 0.5
P ( X ≤ x ) X x Symbol ≤ >
Intuition Continuous cheezon ke liye probabilityarea kyun ban jaati hai
Ek continuous X exactly ek precise number (maan lo exactly 3.0000 … 0 x range mein land karne ki chance," aur woh chance ek curve ke neeche ek area hai.
Topic ko yeh kyun chahiye: yeh ek move — probability = area — poore chapter ka engine hai. Har formula secretly "yeh area compute karo" hi hai.
Definition Function notationf ( x ) function f x f ( x ) f x f ( x )
Definition Probability density function (PDF)Probability density function f ( x ) height tumhe batata hai ki x concentrated hai — zyada height = likely neighbourhood, kam height = rare neighbourhood. Ise legal banane ke do rules hain:
f ( x ) ≥ 0 ( height kabhi negative nahi ) , ∫ − ∞ ∞ f ( x ) d x = 1 ( total area = 1 ) .
f ( x ) Kyun sahi lagta hai: yeh probability aur humhare beech baithta hai. Fix: height density hai (probability per unit length ), probability nahi. Yeh 1 area — height ×
Topic ko yeh kyun chahiye: f ( x ) f
∫ a b f ( x ) d x ∫ a b f ( x ) d x "curve f x = a x = b Pieces padho:
∫ S um ke liye ek stretched "S".d x x f ( x ) d x f ( x ) d x ∫ a b a b
Intuition Yeh tool kyun aur plain multiplication kyun nahi?Agar curve constant height c c × ( b − a ) bend karti hain: height chalte chalte change hoti hai. Multiplication ko constant height chahiye; integral hi ek aisa tool hai jo ek varying height ko sahi se total karta hai, region ko itne patle slivers mein kaatke ki har ek flat ho. Yahi sawal probability poochti hai: "ek curvy shape ke neeche kitna area?"
Topic ko yeh kyun chahiye: P ( a ≤ X ≤ b ) = ∫ a b f ( x ) d x
Definition Cumulative distribution function (CDF)F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t x running total hai. Yeh 0 1 t x
F ′ ( x ) Derivative F ′ ( x ) x slope (steepness) hai — running area wahan kitni tezi se badh raha hai. Tezi se growth matlab bahut density: isliye f ( x ) = F ′ ( x ) slope hai; accumulated area density ka sum hai.
Topic ko yeh kyun chahiye: parent har distribution ko ya toh f F ∫ ′
E [ X ] μ E [ X ] balance point hai — jahan curve ek pivot par balance karti. Hum ise mean kehte hain aur μ μ = E [ X ] = ∫ − ∞ ∞ x f ( x ) d x . f ( x ) d x x
σ Var ( X ) = E [ X 2 ] − μ 2 average squared spread measure karta hai — values mean se average par kitni door, squared, baithi hain. Iska square root σ standard deviation hai, jo X
squared kyun karein?
Mean se plain distance average ho ke 0
Topic ko yeh kyun chahiye: har distribution apne ( μ , σ 2 )
e e − λ t e ≈ 2.718 e − λ t 1 t = 0 t 0 decay karta hai. Jitna bada rate λ
Intuition Waiting problems mein exponentials kyun aate haine − λ t Poisson Process . Isliye yeh tool choose kiya jaata hai kyunki yeh physics se match karta hai, accident se nahi.
Topic ko yeh kyun chahiye: Exponential, Gamma, aur Normal densities sabmein ek e
Definition Symbols jo tumhe milenge
α , β shape parameters; yeh curve ka silhouette mod dete hain.Γ ( α ) Gamma Function , factorial ka ek smooth stand-in: Γ ( n ) = ( n − 1 )! B ( α , β ) Beta Function , woh exact area jo ek Beta curve ko normalize karta hai.Φ N ( 0 , 1 ) π ≈ 3.14159 2 π
Topic ko yeh kyun chahiye: yeh normalising constants hain — woh numbers jinse tum divide karte ho taaki total area exactly 1
Number line and interval a to b
CDF F and derivative F prime
Exponential decay e to minus lambda t
Greek and special functions
Right side cover karo aur dekho ki reveal karne se pehle answer de sakte ho ya nahi.
Ek density ki height f ( x ) nahi hai)? Density = probability per unit length; yeh probability nahi hai. Sirf f f ( x ) 1
Continuous X P ( X = x ) P ( a ≤ X ≤ b ) Ek exact real number hit karne ki chance 0
∫ a b f ( x ) d x "Curve f x = a x = b
f Height vary karti hai; multiplication ko constant height chahiye, isliye hum slivers mein kaatke jo itne patle hoon ki flat lagein aur unhe sum karte hain.
PDF f F F ( x ) = ∫ − ∞ x f ( t ) d t f ( x ) = F ′ ( x )
Mean μ Density ka balance point; μ = ∫ x f ( x ) d x
Variance mein mean se distance ko square kyun karte hain? Taaki overshoots aur undershoots zero tak cancel na ho sakein, aur bade misses zyada count karein.
e − λ t t λ 1 0 λ
Γ B 2 π Φ Yeh normalising constants / area tools hain jo ensure karte hain ki total probability exactly 1 Φ
