Foundations — Chi-squared, t, F distributions — definition, degrees of freedom

4.9.9 · D1 · Maths › Probability Theory & Statistics › Chi-squared, t, F distributions — definition, degrees of fre

Is page mein assume kiya gaya hai ki aap kuch nahi jaante. Hum har ek squiggle ka naam denge jo parent note use karta hai, uske peeche ki picture banayenge, aur batayenge ki yeh topic uske bina kyon nahi chal sakta. Upar se neeche padho — har item upar wale pe lean karta hai.

1. Ek random variable, aur $\sim$ ("distributed as hai")

Wiggle $\sim$ padha jaata hai "is distributed as". Iska matlab "approximately equal" nahi hota. Iska matlab hai "left side ka chance-behaviour right side ki recipe se describe hota hai."

$X\sim N(\mu ,{\sigma }^2)\text{reads}\text{"}X\text{ follows a normal distribution with these settings."}$

Topic ko yeh kyun chahiye: poora chapter ${\chi }^2$, $t$, $F$ ke histogram-shapes ka catalogue hai, aur yeh rule hai ki ek computed quantity kaunsa shape follow karti hai.

2. Normal distribution $N(\mu ,{\sigma }^2)$ aur uske do dials

Figure dekho. Peak $\mu$ par baitta hai. $\mu$ move karo aur poori bell sideways slide karti hai. Centre se "shoulder" (jahan curve bend hoti hai) tak ki distance exactly $\sigma$ hai. Bada $\sigma$ bell ko flatten aur widen karta hai; chhota $\sigma$ use tall aur narrow banata hai.

Topic ko yeh kyun chahiye: parent note sab kuch normals se build karta hai. Use samajhne ke liye hume pehle fluent hona chahiye ki "centre" aur "spread" visually kya matlab rakhte hain.

Normal distribution ke fully-detailed treatment ke liye Standard Normal Distribution dekho.

3. Standard normal $Z$ — single Lego brick

Topic ko yeh kyun chahiye: agar hum hamesha $Z$ mein convert karte hain, to hume ingredient ke taur par sirf ek table / ek shape chahiye. ${\chi }^2$, $t$, $F$ sab "$Z$ ko teen alag tareekon se pakaya gaya" hain.

4. Expectation $E[\cdot ]$ — long-run average

Do facts jinhe hum bahut use karenge:

Standard normal ke liye, $E[Z]=0$ (yeh 0 par centred hai) aur $E[Z^2]=1$ (parent note mein prove kiya gaya hai). Yeh do numbers yaad rakho — yeh chapter ke har mean formula ko seed karte hain.

5. Variance $\mathrm{Var}(\cdot )$ aur independence rule

Doosra form ("square ka mean minus mean ka square") woh workhorse hai jise parent use karta hai $E[Z^2]=\mathrm{Var}(Z)+(E[Z]{)}^2=1$ paane ke liye.

Topic ko yeh kyun chahiye: degrees of freedom independent pieces count karta hai. Independence nahi to clean df nahi.

6. Normal ko square karna: $Z^2$ aur jahan ${\chi }^2$ shuru hota hai

Figure dekho: $Z$ ki symmetric bell (upar) $Z^2$ ki lopsided, hamesha-positive shape banjati hai (neeche). Squaring negative half ko positive half par fold karta hai aur tail ko stretch karta hai. Aise $k$ independent squares ka sum ${\chi }_k^2$ ki definition hai.

Topic ko yeh kyun chahiye: ${\chi }^2$ literally hai squared $Z$'s ka sum. Aap iske definition ko tab tak nahi padh sakte jab tak aap comfortable nahi ho ki $Z^2\ge 0$ hamesha hota hai aur iska skewed shape hai.

7. Degrees of freedom $k$ — free pieces count karna

Memorise karne ka rule: df = (data points ki sankhya) − (unse estimated parameters ki sankhya). Yeh topic mein sabse zyada test kiya jaane wala idea hai.

$n-1$ ki puri kahani ke liye Sample Variance and Bessel's Correction dekho.

8. Sample tools: $n$, $\bar{x}$, $s^2$, aur $\Sigma$

Topic ko yeh kyun chahiye: $t$ aur ${\chi }^2$ practically $\bar{x}$ aur $s^2$ se bane hote hain; $t$ ke exist hone ki poori wajah yahi hai ki hume unknown true $\sigma$ ki jagah random $s$ use karna padta hai.

9. Teen assembled symbols ek nazar mein

Ab har ingredient define ho gayi hai, parent ke headline objects clearly padhte hain:

Fraction bar, square root  ​, aur division yahan ordinary school arithmetic hai — sirf steps 1–8 hi naya content tha. Yahi is page ka point hai: ek baar bricks naam le liye jaayein, buildings simple hain.

Prerequisite map

Har foundation agle mein flow karta hai; bottom node $K$ parent topic hai Chi-squared, t, F distributions — definition, degrees of freedom. Central Limit Theorem — Central Limit Theorem — isliye hai ki $\bar{x}$ pehle se normal kyon hai, aur ${\chi }^2$ shape Gamma Distribution ka special case hai. Yeh distributions phir Hypothesis Testing aur ANOVA ko power karte hain.

Equipment checklist

Right side cover karo; reveal karne se pehle answer de sakte ho?