Continuous Random Variables, CDF and PDF Previous Year Questions (PYQs) for GATE DA: 3+ Solved Questions with Step-by-Step Solutions

Solve 3+ Continuous Random Variables, CDF and PDF previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

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.

Continuous Random Variables, CDF and PDF: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Probability and Statistics MCQ

Consider the cumulative distribution function (CDF) of a random variable

X

F_{X} (x) = ⎩ ⎨ ⎧ 0 \frac{1}{4} (x + 1)^{2} 1 x \leq - 1 - 1 \leq x \leq 1 x \geq 1

The value of

P (X^{2} \leq 0.25)

A.
$0.625$
B.
$0.25$
C.
$0.5$
D.
$0.5625$

Question 2 · Probability and Statistics MCQ

Let

X

be a continuous random variable whose cumulative distribution function (CDF)

F_{X} (x)

, for some

t

, is given as follows:

F_{X} (x) = ⎩ ⎨ ⎧ 0 \frac{x - t}{4 - t} 1 x \leq t t \leq x \leq 4 x \geq 4

If the median of

X

3

, then what is the value of

t

A.
$2$
B.
$1$
C.
$- 1$
D.
$0$

Question 3 · Probability and Statistics MSQ

Let

X

be a discrete valued random variable with cumulative distribution function

F (x)

.

Which of the following statements is/are correct?

A.
$F (x)$ is always a positive function.
B.
$F (x)$ is a non-decreasing function.
C.
$F (x)$ has jump discontinuity.
D.
$F (x)$ is a left continuous function.

A question from this chapter

Question 1

Consider the cumulative distribution function (CDF) of a random variable X : F_{X} (x) = ⎩ ⎨ ⎧ 0 \frac{1}{4} (x + 1)^{2} 1 x \leq - 1 - 1 \leq x \leq 1 x \geq 1 The value of P (X^{2} \leq 0.25) is

Question 2

Let X be a continuous random variable whose cumulative distribution function (CDF) F_{X} (x), for some t, is given as follows: F_{X} (x) = ⎩ ⎨ ⎧ 0 \frac{x - t}{4 - t} 1 x \leq t t \leq x \leq 4 x \geq 4 If the median of X is 3, then what is the value of t ?

Question 3

Let X be a discrete valued random variable with cumulative distribution function F (x) . Which of the following statements is/are correct?

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