Probability Distributions ā Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Probability Distributions, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Probability Distributions in one look
a random variable turns outcomes into numbers, and its distribution says how the probability is spread across them
Random Variables
- a random variable is a real-valued function on the sample space, written X:Sto R
- a capital letter (X, Y, Z) names the random variable itself
- the matching small letter x names one particular value it takes
- the range of X is the set of values X actually assumes
- it can be far smaller than the sample space: ten coins give 1024 outcomes
- but only eleven possible head counts, 0 through 10
- the event [X=x] collects every outcome mapped to that one value x
- P[X=x] is really the probability of that whole collected event
Discrete vs Continuous Random Variable
- possible values are finite or countably infinite
- values are obtained by COUNTING, and are usually non-negative integers
- e.g. the number of heads in ten tosses, or the number of defective items in a sample
- possible values fill a whole interval of real numbers, uncountably infinite
- values are obtained by MEASUREMENT, to whatever precision the instrument allows
- e.g. the height of a tree, or the waiting time for a bus
Probability Distribution: Sum on Two Dice
| x | P(x) |
|---|---|
| 2 | 1/36 |
| 3 | 2/36 |
| 4 | 3/36 |
| 5 | 4/36 |
| 6 | 5/36 |
| 7 | 6/36 |
| 8 | 5/36 |
| 9 | 4/36 |
| 10 | 3/36 |
| 11 | 2/36 |
| 12 | 1/36 |
Probability Mass Function (p.m.f.)
- p_i defined for x_1,x_2,x_3,ā¦, the possible values of X
p_i=P[X=x_i] - every p.m.f. must satisfy both conditions at once
p_iā„0,quad Ī£_i p_i=1
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Probability Distributions sheet covers
- Random Variables
- Discrete vs Continuous Random Variable
- Discrete random variable
- Continuous random variable
- Probability Distribution: Sum on Two Dice
- Probability Mass Function (p.m.f.)
- p_i defined for x_1,x_2,x_3,ā¦, the possible values of X
- every p.m.f. must satisfy both conditions at once
- p.m.f. and c.d.f.: heads in four tosses
- Cumulative Distribution Function (c.d.f.)
- cumulative distribution function of a discrete X
- built from the SAME p.m.f., just accumulated up to x_k
- Solving for the unknown constant in a distribution
- Expected Value and Variance
- mean, or expected value, of X
- variance, the spread around that mean
- the faster computational form, avoids expanding every squared term
- standard deviation, the non-negative square root of the variance
- Special Cases: Degenerate and Uniform Distributions
- Mean and variance of a coin-toss count
- Continuous Random Variable and its p.d.f.
- Reading a density and a cumulative curve
- Verifying a p.d.f. and finding its c.d.f.
- Finding the constant that makes a function a p.d.f.
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Probability Distributions the sheet opens inside the study block itself ā at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.