Probability and Statistics

Syllabus: Mean, median, mode, standard deviation, combinatorial probability, probability distributions, binomial distribution, Poisson distribution, exponential distribution, normal distribution, joint and conditional probability.

5 hand-picked visual resources on this topic, 2 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Conditional probability and Bayes

When a test or a witness is described as accurate some percentage of the time, the base rate usually dominates the answer. A rare event reported by a good detector is still most often a false alarm.

$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$
Bayes
$P(B)=\sum_i P(B|A_i)P(A_i)$

Where marks are lost

Answering with the detector's accuracy and ignoring the prior is exactly the trap these questions are built around.

See it

Standard distributions

Know the mean and variance of each on sight; GATE rarely asks for a derivation.

$\text{Binomial}(n,p):\ \mu=np,\ \sigma^2=np(1-p)$

Mean, variance and expectation

Expectation is linear whether or not the variables are independent; variance adds only when they are uncorrelated.

$E[aX+bY]=aE[X]+bE[Y]$
always, independence not needed

Symmetry arguments

For identically distributed continuous variables, questions like $P(X<Y)$ need no integration — the labels are interchangeable and ties have probability zero.

$X,Y \text{ i.i.d. continuous}\Rightarrow P(X<Y)=\tfrac12$

Median, mode, spread and counting

Mean, median and mode coincide only for a symmetric unimodal distribution — a question that gives all three different is telling you the data is skewed. Standard deviation is almost always faster computed as $\overline{x^2}-\bar x^2$ than from the definition.

← Complex Analysis

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