Which of the following are the properties of Binomial distribution? A. May be symmetrical or skewed B. Uni-modal, bell-shaped and symmetrical C. Asymptotic to the x-axis D. n and p are the two parameters E. μ and σ are the two parameters Choose thecorrectanswer from the options given below:
A and D only
Understanding the Properties of the Binomial Distribution
The Binomial distribution is a fundamental concept in probability theory and statistics. It describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.
Let's analyze the given statements about the properties of the Binomial distribution:
- A. May be symmetrical or skewed: This statement is accurate. The shape of a Binomial distribution depends on the probability of success (p). If \(p = 0.5\), the distribution is symmetrical. If \(p < 0.5\), it is skewed to the right. If \(p > 0.5\), it is skewed to the left.
- B. Uni-modal, bell-shaped and symmetrical: This description primarily fits the Normal distribution, especially when the number of trials (n) is large and p is close to 0.5. A general Binomial distribution is not always bell-shaped and symmetrical. Thus, this statement is not a universal property of the Binomial distribution.
- C. Asymptotic to the x-axis: Asymptotic behavior (approaching the x-axis but never touching it) is characteristic of continuous probability density functions like the Normal distribution, where the probability is defined over a continuous range and the tails extend infinitely. The Binomial distribution is a discrete probability distribution, meaning it only has probabilities for specific integer values (number of successes). It is not asymptotic to the x-axis.
- D. n and p are the two parameters: This statement is correct. The Binomial distribution is defined by two parameters: 'n', which represents the number of independent Bernoulli trials, and 'p', which represents the probability of success on any single trial. The probability mass function (PMF) of a Binomial distribution depends directly on these two values: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\).
- E. μ and σ are the two parameters: This statement is incorrect. \(\mu\) (mean) and \(\sigma\) (standard deviation) are the parameters for the Normal distribution. While the mean and standard deviation of a Binomial distribution can be calculated from its parameters n and p (\(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\)), \(\mu\) and \(\sigma\) themselves are not the defining parameters of the Binomial distribution.
Based on the analysis, the properties that accurately describe the Binomial distribution are A and D.
Identifying Key Binomial Distribution Properties
The Binomial distribution is characterized by several key features:
- It is a discrete probability distribution.
- It models the number of successes in a fixed number of trials (n).
- Each trial is independent.
- There are only two possible outcomes for each trial (success or failure).
- The probability of success (p) is constant for each trial.
- Its shape can be symmetrical or skewed, depending on the value of p.
- It is defined by the parameters n (number of trials) and p (probability of success).
Revision Table: Binomial Distribution Properties
| Property | Applicability to Binomial Distribution | Explanation |
|---|---|---|
| May be symmetrical or skewed | Yes | Shape depends on p (probability of success). Symmetrical if p=0.5, skewed otherwise. |
| Uni-modal, bell-shaped, symmetrical | No (Generally) | This typically describes the Normal distribution. Binomial is only approximately so for large n, p ≈ 0.5. |
| Asymptotic to the x-axis | No | Discrete distribution, probabilities are only defined for specific points, not a continuous curve. |
| n and p are parameters | Yes | These define the distribution and its probability mass function. |
| \(\mu\) and \(\sigma\) are parameters | No | These are parameters for the Normal distribution. Mean (\(\mu = np\)) and standard deviation (\(\sigma = \sqrt{np(1-p)}\)) can be derived from n and p. |
Additional Information on Probability Distributions
Understanding different types of probability distributions is crucial in statistics. Here's how the Binomial distribution relates to other concepts:
- Bernoulli Trial: A single trial with exactly two outcomes (success/failure) and a fixed probability of success (p) is called a Bernoulli trial. The Binomial distribution is the sum of n independent Bernoulli trials.
- Normal Distribution: This is a continuous, symmetrical, bell-shaped distribution defined by parameters \(\mu\) (mean) and \(\sigma\) (standard deviation). The Normal distribution can be used to approximate the Binomial distribution when n is large and p is not too close to 0 or 1 (rule of thumb: \(np \geq 5\) and \(n(1-p) \geq 5\)).
- Parameters: Parameters are values that define a specific probability distribution. For the Binomial distribution, these are the number of trials (n) and the probability of success (p). For the Normal distribution, these are the mean (\(\mu\)) and standard deviation (\(\sigma\)).
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:
