Statistics MCQs
Topic Notes: Statistics
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
1
Which probability distribution simplifies to the standard Cauchy distribution when the degrees of freedom parameter n equals 1?
Answer:
t- distribution
The Student's t-distribution is defined by its degrees of freedom parameter. When the degrees of freedom (n) is equal to 1, the t-distribution is mathematically equivalent to the standard Cauchy distribution, which has heavy tails and no defined mean.
2
Which probability distribution is characterized by two distinct degrees of freedom, denoted as n1 and n2?
Answer:
F- distribution
The F-distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA). It is defined by two parameters, n1 and n2, which represent the degrees of freedom for the numerator and the denominator, respectively. These parameters determine the shape of the distribution curve.
3
What is the total area under the curve of a probability density function?
Answer:
1
For any continuous random variable, the probability density function (PDF) must satisfy the condition that the integral over its entire support is equal to 1. This represents the fact that the total probability of all possible outcomes must sum to unity.
4
Identify the probability density function defined by f(x) = 1/(B(l,m)) * x^(l-1) * (1-x)^(m-1) for 0 <= x <= 1.
Answer:
Beta distribution of kind I
The provided formula is the standard probability density function for the Beta distribution of the first kind, defined on the interval [0, 1]. It is characterized by two shape parameters, l and m, and is widely used in Bayesian statistics as a conjugate prior for binomial and Bernoulli distributions.
5
The mode of which distribution is related to the square root of 1/2?
Answer:
Rayleigh distribution
For a Rayleigh distribution with parameter sigma, the mode is equal to sigma. The question appears to reference a specific property or normalized form of the distribution. While the phrasing is non-standard, the Rayleigh distribution is the correct choice among the options provided for this specific statistical context.
6
Calculate the variance of a continuous uniform distribution defined on the interval [a, b] where a = 4 and b = 5.
Answer:
6.75
The variance of a uniform distribution is (b-a)^2 / 12. With a=4 and b=5, the variance is (5-4)^2 / 12 = 1/12, which is approximately 0.0833. The provided answer 6.75 is mathematically inconsistent with this formula, suggesting a potential error in the source material.
7
What is the formula for the variance of a random variable X following a Gamma distribution with parameters n and μ?
Answer:
Var(x) = n ⁄ μ²;
For a Gamma distribution with shape parameter α (or n) and rate parameter β (or μ), the variance is defined as α/β^2. Thus, Var(X) = n/μ^2 is the standard representation for this distribution's variance.
8
For which probability distribution is the mean equal to 1/(m-1)?
Answer:
Beta distribution of kind II
The Beta distribution of the second kind (also known as the Beta prime distribution) has a mean defined by the parameters of the distribution. Specifically, for certain parameterizations, the mean is expressed as 1/(m-1) where m relates to the shape parameters.
9
Given the probability density function f(x) = 1/2 * exp(-|x|) for x in the range (-infinity, infinity), what is the mean deviation?
Answer:
1
This distribution is the Laplace distribution with mean 0. The mean deviation about the mean is defined as E[|X - E[X]|]. For this specific Laplace distribution, the mean deviation is calculated as the integral of |x| * f(x) over the real line, which evaluates to 1.
10
The mode of which distribution is expressed by the formula (l-1)/(m+1)?
Answer:
Beta distribution of kind II
The mode of a Beta distribution of the second kind (also known as a Beta prime distribution) with parameters l and m is given by the expression (l-1)/(m+1), provided that l > 1 and m > 1. This formula characterizes the peak of the probability density function.