Discrete Distribution Models Explained
A discrete probability distribution is used when the outcomes of a random variable can only take specific, separate values, usually integers. These are often counts. We need to identify which of the listed options is not a recognized discrete probability distribution model.
Binomial Distribution Overview
The Binomial distribution is a common discrete distribution. It calculates the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (like success or failure). For example, flipping a coin 10 times and finding the probability of getting exactly 7 heads.
Negative Binomial Distribution
The Negative binomial distribution is also a discrete probability distribution. It models the number of trials needed to achieve a specified number of successes. For instance, it could calculate the probability that you need 15 coin flips to get your 5th head.
Poisson Distribution Characteristics
The Poisson distribution is another fundamental discrete distribution. It's used to predict the number of events occurring within a specific time or space interval, given a known average rate. An example is predicting the number of customers arriving at a store in one hour, assuming an average arrival rate.
Positive Binomial Uncertainty
The term "Positive binomial" is not a standard, established name for a discrete probability distribution model in statistics. While the outcomes of a standard binomial distribution are non-negative integers (fitting a broad sense of 'positive' counts), there isn't a distinct model formally known or widely used by this specific name in probability theory textbooks or common practice, unlike the Binomial, Negative Binomial, and Poisson distributions.
Distribution Model Comparison
Comparing the options, the Binomial, Negative Binomial, and Poisson distributions are all well-defined and frequently used discrete probability models. The "Positive binomial" does not fit this category as a standard named distribution.
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:
