Probability Distributions & Random Variables
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📘 Comprehensive Syllabus & Examination Guide
Probability Distributions & Random Variables
Official curriculum roadmap, subject/topic distribution, negative marking rules, pacing guidelines, and solved sample questions.
🎯 Mapped Subjects & Topic Question Distribution
Total Question Pool
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90 MCQs
Combined Active Syllabus
Probability Distributions & Random Variables
90 MCQs
Topic Pool
📊 Question Pool Structure
90 MCQs across fundamental, intermediate, and advanced concept tiers.
⚡ Recommended Pacing
45 to 60 seconds per MCQ. Flag complex problems and preserve 10 minutes for final revision.
⚖️ Scoring & Negative Marking
+1 mark per correct answer. In competitive tests with negative marking, -0.25 applies for incorrect guesses.
💡 Strategic Preparation & Exam Hall Guidelines
To maximize your score on Probability Distributions & Random Variables, candidates are advised to follow a structured three-pass approach. In the First Pass, solve all direct recall and formula-based questions within 30 seconds each to secure foundational marks. In the Second Pass, tackle multi-step analytical and quantitative reasoning problems. In the Third Pass, review marked questions and verify calculations.
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📝 Pre-Rendered Solved Sample Questions & Detailed Solutions
Review the solved problems below to understand question phrasing, answer choices, and step-by-step solution logic prior to starting the full interactive practice drill:
Sample Question 1
Probability Distributions & Random Variables
medium • Statistics
If a random variable $X$ follows a Poisson distribution with parameter $\lambda = 4$, what is the coefficient of skewness $\gamma_1 = \frac{\mu_3}{\sigma^3}$?
A
0.25
B
0.50
C
1.00
D
2.00
✓ Correct Answer: B - 0.50
📖 Step-by-Step Solution & Conceptual Rationale:
For a Poisson distribution with parameter $\lambda$, the variance $\sigma^2 = \lambda$ and the third central moment $\mu_3 = \lambda$. Therefore, skewness $\gamma_1 = \frac{\lambda}{\lambda^{3/2}} = \frac{1}{\sqrt{\lambda}} = \frac{1}{\sqrt{4}} = 0.5$.
Sample Question 2
Probability Distributions & Random Variables
easy • Statistics
For a Binomial distribution $B(n, p)$, under what condition does the distribution become perfectly symmetrical?
A
When $n \to \infty$ for any $p$
B
When $p = 0.5$
C
When $n p = 1$
D
When $p = 0.25$
✓ Correct Answer: B - When $p = 0.5$
📖 Step-by-Step Solution & Conceptual Rationale:
The Binomial distribution is perfectly symmetrical (skewness $\gamma_1 = \frac{q-p}{\sqrt{npq}} = 0$) if and only if $p = q = 0.5$, regardless of the sample size $n$.
Sample Question 3
Probability Distributions & Random Variables
medium • Statistics
What is the Moment Generating Function (MGF) $M_X(t)$ of an Exponential random variable $X \sim \text{Exp}(\lambda)$ with probability density function $f(x) = \lambda e^{-\lambda x}$ for $x > 0$ and $t < \lambda$?
A
$\frac{\lambda}{\lambda - t}$
B
$\frac{\lambda}{\lambda + t}$
C
$e^{\lambda t}$
D
$\frac{1}{1 - \lambda t^2}$
✓ Correct Answer: A - $\frac{\lambda}{\lambda - t}$
📖 Step-by-Step Solution & Conceptual Rationale:
The MGF is $M_X(t) = \int_0^\infty e^{tx} \lambda e^{-\lambda x} dx = \lambda \int_0^\infty e^{-(\lambda - t)x} dx = \frac{\lambda}{\lambda - t}$ for $t < \lambda$.
Sample Question 4
Probability Distributions & Random Variables
easy • Statistics
If $X$ and $Y$ are two independent standard normal random variables ($X, Y \sim N(0, 1)$), what is the probability distribution of $Z = X^2 + Y^2$?
A
Standard Normal distribution $N(0, 2)$
B
Chi-Square distribution with 2 degrees of freedom $\chi^2_{(2)}$ (or Exponential with mean 2)
C
Student's t-distribution with 2 degrees of freedom
D
F-distribution with $(1, 1)$ degrees of freedom
✓ Correct Answer: B - Chi-Square distribution with 2 degrees of freedom $\chi^2_{(2)}$ (or Exponential with mean 2)
📖 Step-by-Step Solution & Conceptual Rationale:
The sum of squares of $k$ independent standard normal variables follows a Chi-square distribution with $k$ degrees of freedom. For $k=2$, $\chi^2_{(2)}$ is identical to an Exponential distribution with parameter $\lambda = 1/2$ (mean 2).
Sample Question 5
Probability Distributions & Random Variables
easy • Statistics
Which probability distribution possesses the unique 'Memoryless Property' among continuous random variables?
A
Normal distribution
B
Exponential distribution
C
Uniform distribution
D
Weibull distribution with shape parameter $k = 2$
✓ Correct Answer: B - Exponential distribution
📖 Step-by-Step Solution & Conceptual Rationale:
The Exponential distribution satisfies $P(X > s + t \mid X > s) = P(X > t)$, meaning the past duration has no effect on future probability; it is the only continuous distribution with this property.
Sample Question 6
Probability Distributions & Random Variables
easy • Statistics
Which discrete probability distribution possesses the memoryless property ($P(X > s + t \mid X > s) = P(X > t)$)?
A
Binomial distribution
B
Geometric distribution
C
Poisson distribution
D
Hypergeometric distribution
✓ Correct Answer: B - Geometric distribution
📖 Step-by-Step Solution & Conceptual Rationale:
The Geometric distribution is the sole discrete distribution that is memoryless, modeling the number of Bernoulli trials until the first success.
Sample Question 7
Probability Distributions & Random Variables
medium • Statistics
If a random variable $X$ has a standard Cauchy distribution ($f(x) = \frac{1}{\pi(1+x^2)}$), what is its mathematical expectation $E[X]$?
A
0
B
Undefined (does not exist)
C
1
D
$\pi$
✓ Correct Answer: B - Undefined (does not exist)
📖 Step-by-Step Solution & Conceptual Rationale:
The integral $\int_{-\infty}^\infty \frac{x}{\pi(1+x^2)} dx$ is not absolutely convergent (diverges to $\infty - \infty$). Hence, the mean and all higher moments of the Cauchy distribution do not exist.
Sample Question 8
Probability Distributions & Random Variables
easy • Statistics
If $X_1, X_2, \dots, X_n$ are independent random variables from a Poisson distribution with parameters $\lambda_1, \lambda_2, \dots, \lambda_n$, what is the distribution of their sum $S_n = \sum_{i=1}^n X_i$?
A
Normal distribution with mean $\sum \lambda_i$
B
Poisson distribution with parameter $\lambda = \sum_{i=1}^n \lambda_i$
C
Binomial distribution with $p = \sum \lambda_i / n$
D
Negative Binomial distribution
✓ Correct Answer: B - Poisson distribution with parameter $\lambda = \sum_{i=1}^n \lambda_i$
📖 Step-by-Step Solution & Conceptual Rationale:
By the additive (reproductive) property of the Poisson distribution, the sum of independent Poisson variables is itself Poisson distributed with parameter equal to the sum of their individual means.
Sample Question 9
Probability Distributions & Random Variables
easy • Statistics
What is the variance of a Continuous Uniform distribution $U(a, b)$ on the interval $[a, b]$?
A
$\frac{(b - a)^2}{12}$
B
$\frac{(b - a)^2}{6}$
C
$\frac{b - a}{2}$
D
$\frac{(b + a)^2}{12}$
✓ Correct Answer: A - $\frac{(b - a)^2}{12}$
📖 Step-by-Step Solution & Conceptual Rationale:
For a uniform distribution $U(a, b)$, the mean is $\mu = \frac{a+b}{2}$ and the variance is $\sigma^2 = \frac{(b-a)^2}{12}$.
Sample Question 10
Probability Distributions & Random Variables
medium • Statistics
If $X \sim N(\mu, \sigma^2)$, what are the values of the third and fourth central moments $\mu_3$ and $\mu_4$?
A
$\mu_3 = 0$ and $\mu_4 = 3\sigma^4$
B
$\mu_3 = \sigma^3$ and $\mu_4 = \sigma^4$
C
$\mu_3 = 0$ and $\mu_4 = 0$
D
$\mu_3 = 3\sigma^3$ and $\mu_4 = 3\sigma^4$
✓ Correct Answer: A - $\mu_3 = 0$ and $\mu_4 = 3\sigma^4$
📖 Step-by-Step Solution & Conceptual Rationale:
Due to symmetry around $\mu$, all odd central moments of the normal distribution are zero ($\mu_3 = 0$). The fourth central moment is $\mu_4 = 3\sigma^4$, giving a kurtosis coefficient $\beta_2 = \mu_4 / \sigma^4 = 3$ (mesokurtic).
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