Question

Consider the following distributions:
(A). Normal distribution
(B). Binomial distribution
(C). Poisson distribution
(D). F-distribution
(E). Chi-square distribution
The distributions which are of continuous nature are:

• A. (A), (B) and (C) only.
• B. (B), (D) and (E) only.
• C. (A), (B), (C) and (E) only.
• D. (A), (D) and (E) only.

Hint:

To distinguish between discrete and continuous distributions, ask yourself: "Does this variable count occurrences or measure physical properties" Counting processes (like counts of successes or events) yield discrete variables, while measurements (like time, ratios, or values along an interval) yield continuous variables.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Probability distributions are broadly classified into two categories based on the sample space of the underlying random variable: discrete distributions and continuous distributions.

Step 2: Detailed Explanation:
Let us classify each of the given distributions:

(A) Normal distribution: This is a continuous distribution defined over the real number line, from \(-\infty\) to \(+\infty\). It is characterized by its bell-shaped curve and is defined by its mean (\(\mu\)) and variance (\(\sigma^2\)).

(B) Binomial distribution: This is a discrete distribution that models the number of successes in a fixed number \(n\) of independent trials, where each trial has only two possible outcomes (success or failure). The random variable can only take discrete integer values in the range \([0, n]\).

(C) Poisson distribution: This is another discrete distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. The random variable representing the count of events can only take non-negative integer values \(\{0, 1, 2, \dots\}\).

(D) F-distribution: This is a continuous distribution that arises in the analysis of variance (ANOVA). It is defined as the ratio of two independent chi-square variables, each divided by its respective degrees of freedom, and takes any non-negative real value in the interval \([0, \infty)\).

(E) Chi-square (\(\chi^2\)) distribution: This is a continuous distribution representing the sum of the squares of \(k\) independent, standard normal random variables. It takes continuous values in the range \([0, \infty)\).
Thus, the continuous distributions are (A), (D), and (E).

Step 3: Final Answer:
The continuous distributions are (A), (D), and (E), which corresponds to Option (D).