Continuous Probability and Geometric Probability Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuous Probability and Geometric Probability notes for GATE DA: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Continuous Probability and Geometric Probability

Your journey through this chapter

Continuous Random Variables and PDFs

From counting discrete outcomes to measuring continuous ones.

The Continuous Uniform Distribution

The simplest continuous model, where every interval of the same length is equally likely.

Geometric Probability

When chance meets geometry: lengths, areas, and volumes.

Joint Distributions and Area Methods

Solving multi-variable problems by calculating areas in the plane.

End goal: visualize probability as area under a curve or as a ratio of geometric measures, and solve continuous probability questions effortlessly.

The Shift to Continuous: From Counting to Measuring

The fundamental shift

Discrete: Outcomes are countable. We use a Probability Mass Function (PMF). $P (X = x) > 0$ .
Continuous: Outcomes are measurements (time, length, area). We use a Probability Density Function (PDF). $P (X = x) = 0$ .

The Probability Density Function (PDF)

For a continuous random variable $X$ , the PDF $f (x)$ satisfies:

$f (x) \geq 0$ for all $x$ .
The total area under the curve is 1: $\int_{- \infty}^{\infty} f (x) d x = 1$ .

How to find probabilities

The probability that $X$ falls in an interval $[a, b]$ is the area under the PDF over that interval:

P (a \leq X \leq b) = \int_{a}^{b} f (x) d x

Key intuition: In continuous probability, you do not calculate the probability of a point; you calculate the probability of a region by measuring its area.

The Continuous Uniform Distribution

Definition

A continuous random variable $X$ has a uniform distribution on the interval $[a, b]$ , written as $X \sim U (a, b)$ , if its PDF is constant over $[a, b]$ and zero elsewhere.

The PDF

f (x) = ⎩ ⎨ ⎧ \frac{1}{b - a} 0 for a \leq x \leq b otherwise

Visualizing the PDF

The graph of $f (x)$ is a horizontal rectangle.

The base is the interval $[a, b]$ , which has length $b - a$ .
The height is $\frac{1}{b - a}$ .
Area = base $\times$ height = $(b - a) \times \frac{1}{b - a} = 1$ .

Core property For any sub-interval

[c, d]

inside

[a, b]

P (c \leq X \leq d) = \frac{d - c}{b - a}

The probability depends only on the length of the interval, not its location.

Mean, Variance, and CDF of the Uniform Distribution

Mean (Expected Value)

The center of the interval:

E [X] = \frac{a + b}{2}

Variance

Depends only on the width of the interval:

Var (X) = \frac{( b - a ) ^{2}}{12}

Cumulative Distribution Function (CDF)

The CDF $F (x) = P (X \leq x)$ accumulates area from the left:

F (x) = ⎩ ⎨ ⎧ 0 \frac{x - a}{b - a} 1 for x < a for a \leq x \leq b for x > b

Memory hook: The CDF is just the fraction of the interval

[a, b]

that lies to the left of

x

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Continuous Probability and Geometric Probability Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuous Probability and Geometric Probability notes for GATE DA: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

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