Continuous Random Variables, CDF and PDF Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuous Random Variables, CDF and PDF notes for GATE DA: 10 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: CDF Evaluation, Properties and Quantiles

Chapter Roadmap

Chapter: Continuous Random Variables, CDF and PDF. Topic: CDF Evaluation, Distribution Properties and Quantiles.

Meaning of the CDF

Accumulated probability

F (x) = P (X \leq x)

Validity properties

Limits, monotonicity, right-continuity, jumps.

Probabilities from the CDF

Intervals, tails, and endpoint rules.

Transformed events

Rewrite events like

X^{2} \leq c

into intervals for

X

Quantiles and median

Invert the CDF to find cutoff values.

Common traps

Discrete jumps, strict inequalities, piecewise boundaries.

Weightage hint: high. This is the first topic of the chapter and a core dependency for later continuous probability ideas.

Topic Hero: The CDF Is an Accumulator of Probability

The CDF Is an Accumulator

Sweep right: collect probability mass

F(x)

F_{X} (x) = P (X \leq x) .

What fraction of the probability mass lies at or below $x$ ?

For a continuous random variable, the probability of one exact point is zero:

P (X = x) = 0.

Far to the left, no probability has been accumulated: $F (- \infty) = 0$ .
As $x$ moves right, more probability is collected.
Far to the right, all probability has been collected: $F (+ \infty) = 1$ .
The CDF never falls. It either rises or stays flat.

Properties Every Valid CDF Must Satisfy

Required CDF Properties

Property	Condition	Meaning
Lower limit	$x \to - \infty lim F (x) = 0$	No probability far to the left
Upper limit	$x \to + \infty lim F (x) = 1$	Total probability is one
Non-decreasing	$x_{1} < x_{2} ⟹ F (x_{1}) \leq F (x_{2})$	Accumulated probability cannot fall
Right-continuity	$h \to 0^{+} lim F (x + h) = F (x)$	The CDF value matches the limit from the right

Left limit

F (x -) = h \to 0^{+} lim F (x - h) = P (X < x) .

Jump size

F (x) - F (x -) = P (X = x) .

For a purely continuous random variable, there are no jumps, so

F (x -) = F (x)

Extracting Probabilities from a CDF

Probability Extraction

P (a < X \leq b) = F (b) - F (a) .

For a continuous random variable, point probabilities are zero, so endpoints do not matter:

P (a < X < b) = P (a \leq X \leq b) = P (a < X \leq b) = F (b) - F (a) .

Target probability	General CDF form	Continuous simplification
$P (X \leq k)$	$F (k)$	$F (k)$
$P (X > k)$	$1 - F (k)$	$1 - F (k)$
$P (X < k)$	$F (k -)$	$F (k)$
$P (X \geq k)$	$1 - F (k -)$	$1 - F (k)$

For a continuous variable and

k \geq 0

P (∣ X ∣ \leq k) = P (- k \leq X \leq k) = F (k) - F (- k) .

Always identify the correct piece of a piecewise CDF before substituting.

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Continuous Random Variables, CDF and PDF Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuous Random Variables, CDF and PDF notes for GATE DA: 10 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questio

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