Which of the following are the properties of normal distribution? (A) May be symmetrical or skewed (B) Uni-modal, bell-shaped and symmetrical (C) Asymptotic to the x-axis (D) m and p are the two parameters (E) μ and σ are the two parameters Choose the correct answer from the options given below:
(B), (C), (E) Only
Understanding Normal Distribution Properties
The question asks about the key properties of a normal distribution. The normal distribution is a very common and important probability distribution in statistics. It is also known as the Gaussian distribution. Let's analyze each of the given statements to determine which ones are true properties of the normal distribution.
Analyzing Each Property of Normal Distribution
- (A) May be symmetrical or skewed: A normal distribution is always perfectly symmetrical. Its shape is defined such that the left side is a mirror image of the right side. If a distribution is skewed (either positively or negatively), it is not a normal distribution. Therefore, this statement is incorrect.
- (B) Uni-modal, bell-shaped and symmetrical: This is a classic description of the normal distribution curve.
- Uni-modal: It has a single peak.
- Bell-shaped: The curve resembles the shape of a bell.
- Symmetrical: As mentioned before, it is symmetrical about its mean.
- (C) Asymptotic to the x-axis: The tails of the normal distribution curve extend infinitely in both directions, approaching the x-axis but never actually touching it. This property means the probability density is always positive, though extremely small far from the mean. This statement is correct.
- (D) m and p are the two parameters: The parameters 'm' and 'p' are not the standard parameters used to define a normal distribution. Parameters 'n' and 'p' are typically associated with the binomial distribution, representing the number of trials and the probability of success, respectively. The normal distribution is defined by different parameters. This statement is incorrect.
- (E) $\mu$ and $\sigma$ are the two parameters: The normal distribution is completely determined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The mean ($\mu$) determines the center of the distribution, and the standard deviation ($\sigma$) determines the spread or width of the distribution. This statement is correct.
Identifying Correct Properties of Normal Distribution
Based on our analysis, the true properties of a normal distribution among the given options are (B), (C), and (E). Therefore, we look for the option that includes only these statements.
Conclusion on Normal Distribution Properties
The statements that correctly describe the properties of a normal distribution are:
- (B) Uni-modal, bell-shaped and symmetrical
- (C) Asymptotic to the x-axis
- (E) $\mu$ and $\sigma$ are the two parameters
The option that lists (B), (C), and (E) only is the correct answer.
| Property | Statement | Correct? | Explanation |
|---|---|---|---|
| A | May be symmetrical or skewed | No | Normal distribution is always symmetrical. |
| B | Uni-modal, bell-shaped and symmetrical | Yes | Defines the characteristic shape. |
| C | Asymptotic to the x-axis | Yes | Tails approach but never touch the axis. |
| D | m and p are the two parameters | No | Parameters are $\mu$ and $\sigma$. 'n' and 'p' are for binomial. |
| E | $\mu$ and $\sigma$ are the two parameters | Yes | These define the center and spread. |
Revision Table: Key Normal Distribution Characteristics
| Characteristic | Description |
|---|---|
| Shape | Bell-shaped and Symmetrical |
| Modality | Uni-modal (single peak) |
| Center | Mean ($\mu$), Median, and Mode are all equal and located at the center. |
| Spread | Determined by the Standard Deviation ($\sigma$). |
| Tails | Asymptotic to the x-axis (extend infinitely without touching). |
| Parameters | Mean ($\mu$) and Standard Deviation ($\sigma$). |
| Total Area | The total area under the curve is equal to 1 (or 100%). |
Additional Information about Normal Distribution
The normal distribution is crucial because many natural phenomena and measurements tend to follow this pattern (e.g., heights, test scores, measurement errors). It is also fundamental in statistical inference.
- Empirical Rule (68-95-99.7 Rule): For a normal distribution, approximately:
- 68% of the data falls within one standard deviation ($\pm 1\sigma$) of the mean ($\mu$).
- 95% of the data falls within two standard deviations ($\pm 2\sigma$) of the mean ($\mu$).
- 99.7% of the data falls within three standard deviations ($\pm 3\sigma$) of the mean ($\mu$).
- Standard Normal Distribution: A special case where the mean ($\mu$) is 0 and the standard deviation ($\sigma$) is 1. Any normal distribution can be transformed into a standard normal distribution using the z-score formula: $Z = \frac{X - \mu}{\sigma}$.
- Continuous Probability Distribution: The normal distribution is a continuous probability distribution, meaning it describes probabilities for continuous random variables.
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:
