Match List - I with List - II : Choose the correct answer from the options given below :List - I (Description) List - II (Term) A. The probability of both success and failure remains constant I. Binomial Distribution B. The mean of distribution may be negative or positive II. Poisson Distribution C. The sum of all probabilities is equal to 1 III. Normal Distribution D. The probability of occurrence of an outcome within a very small time period is very small IV. Random Variable
This question requires matching descriptions of probability concepts with their corresponding terms. Let's break down each item in List - I and find the best fit from List - II.
A. Matching Binomial Distribution Characteristics
The description "The probability of both success and failure remains constant" perfectly describes the conditions for a Binomial Distribution (List - I, Item I). In a binomial setting, we have a fixed number of independent trials, and each trial has only two possible outcomes (success or failure) with a constant probability of success, denoted as '$p$', and a constant probability of failure, denoted as '$1-p$'. This constancy is a defining feature.
B. Matching Normal Distribution Mean Property
The statement "The mean of distribution may be negative or positive" aligns with the properties of a Normal Distribution (List - I, Item III). The mean, often represented by the Greek letter '$\mu$', determines the center of the bell-shaped curve. $\mu$ can take any real value, meaning it can be positive, negative, or zero. While other distributions might have restricted means (e.g., Poisson mean is non-negative), the normal distribution's mean is unrestricted.
C. Matching Sum of Probabilities Property
The principle that "The sum of all probabilities is equal to 1" is a fundamental rule for any probability distribution. For a discrete Random Variable (List - I, Item IV), the sum of the probabilities of all its possible values must equal 1. Although this property applies broadly to all distributions (Binomial, Poisson, Normal), in the context of the provided options and the required matching, 'Random Variable' is the designated match. It implies summing the probabilities across all possible outcomes that the random variable can take.
Key Principle: For any random variable '$X$', the sum over all possible values '$x$' is:
$$ \sum P(X=x) = 1 $$
D. Matching Poisson Distribution Scenario
The characteristic "The probability of occurrence of an outcome within a very small time period is very small" is a hallmark of the Poisson Distribution (List - I, Item II). The Poisson distribution is used to model the number of events occurring within a specific interval of time or space, given a constant average rate of occurrence. When the interval is made very small, the likelihood of multiple events happening is negligible, and the probability of a single event occurring becomes very small, aligning with the distribution's assumptions.
Summary of Matches
Based on the analysis of the characteristics of each probability term:
| List - I Description | List - II Term | Reasoning |
| A. Constant probability of success/failure | I. Binomial Distribution | Defines the binomial trial conditions. |
| B. Mean can be negative or positive | III. Normal Distribution | The mean ($\mu$) of a normal distribution can be any real number. |
| C. Sum of all probabilities is 1 | IV. Random Variable | Refers to the sum of probabilities across all possible outcomes of a random variable. |
| D. Small probability in a small time period | II. Poisson Distribution | Describes events occurring at a constant rate over time/space. |
Therefore, the correct matching is A-I, B-III, C-IV, D-II.
Important Questions from Probability Distribution
If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:
For the distribution with unknown θ
\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)
We set the testing of hypothesis H 0 ∶ θ = 1 vs H 1 ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:
For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)
the upper quartile point is
Let the joint probability density function of \( (X, Y) \) be
\[f(x, y) = \begin{cases} 6xy^2 & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}\]
Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is:
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
