Discrete Random Variables and Bernoulli-Binomial Models Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Discrete Random Variables and Bernoulli-Binomial Models notes for GATE DA: 20 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Bernoulli and Binomial Models

Chapter Roadmap

Bernoulli Trial

Model one yes/no outcome with success probability

p

Binomial Count

Repeat independent trials and count successes.

Exact Probabilities & Expectations

Use PMF, complements, and linearity of expectation.

Normal Approximation

Approximate large-

n

counts using mean and SD.

Weightage hint: High. This topic is central for repeated independent counting models. By the end, you can compute exact binomial probabilities, expected counts, and normal approximations.

Topic Hero: One Yes/No Trial Becomes a Counting Model

Topic Hero

Many random experiments reduce to the same structure: repeated yes/no trials where we count how often one side occurs.

Situation	Success	Failure
Die thrown	Shows 1	Does not show 1
Ball drawn (w/ replacement)	Black ball	White ball
Matrix entry generated	Entry is 1	Entry is 0

In each case, the real target is not one isolated outcome. The target is the count of successes over many independent repetitions. Once you see that structure, the binomial model becomes the natural tool.

Bernoulli Random Variable: The Building Block

Bernoulli Random Variable

X \sim Bernoulli (p)

SUCCESS

X = 1

P (X = 1) = p

FAILURE

X = 0

P (X = 0) = 1 - p

Mean:

E [X] = p

Variance:

Var (X) = p (1 - p)

Indicator Variable: If

A

is an event,

I (A) = 1

A

occurs, else

0

. This is exactly a Bernoulli variable with

p = P (A)

From Bernoulli Trials to Binomial Count

Binomial Distribution

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent $Bernoulli (p)$ variables. Define $Y = \sum_{i = 1}^{n} X_{i}$ . Then:

Y \sim Binomial (n, p)

Probability Mass Function

P (Y = k) = (k n) p^{k} (1 - p)^{n - k}

for

k = 0, 1, 2, \dots, n

Mean

E [Y] = n p

Variance

Var (Y) = n p (1 - p)

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Discrete Random Variables and Bernoulli-Binomial Models Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Discrete Random Variables and Bernoulli-Binomial Models notes for GATE DA: 20 study cards covering concepts, formulas, shortcuts and exam traps, plus solved p

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