The number of parameters in the univariate exponential and Gaussian distributions, respectively are
1 and 2
Understanding the number of parameters in probability distributions is fundamental in statistics and probability theory. Each distribution is defined by a set of parameters that dictate its shape, location, and scale. Let's analyze the univariate exponential distribution and the univariate Gaussian distribution to determine their respective numbers of parameters.
Exponential Distribution Parameters
The univariate exponential distribution is often used to model the time until an event occurs in a Poisson process, where events occur continuously and independently at a constant average rate. Its behavior is entirely determined by a single parameter.
- Probability Density Function (PDF): The PDF of a univariate exponential distribution is given by: $$f(x; \lambda) = \lambda e^{-\lambda x} \quad \text{for } x \ge 0$$ Where $x$ represents the random variable (e.g., time), and $\lambda$ (lambda) is the rate parameter.
- Rate Parameter ($\lambda$): This parameter represents the average number of events per unit of time or the inverse of the mean time between events. A larger $\lambda$ means events occur more frequently.
- Number of Parameters: As evident from its PDF, the univariate exponential distribution has 1 parameter, which is $\lambda$.
Gaussian Distribution Parameters
The univariate Gaussian distribution, also known as the normal distribution, is one of the most important continuous probability distributions in statistics. It is characterized by its symmetric, bell-shaped curve and is defined by two key parameters.
- Probability Density Function (PDF): The PDF of a univariate Gaussian distribution is given by: $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \quad \text{for } x \in \mathbb{R}$$ Where $x$ represents the random variable, $\mu$ (mu) is the mean parameter, and $\sigma^2$ (sigma squared) is the variance parameter.
- Mean Parameter ($\mu$): This parameter determines the central tendency or the peak of the distribution. It represents the average value of the data.
- Variance Parameter ($\sigma^2$): This parameter measures the spread or dispersion of the data around the mean. A larger $\sigma^2$ indicates wider spread data, while a smaller $\sigma^2$ indicates data clustered more closely around the mean. Alternatively, the standard deviation ($\sigma$) can be used instead of variance ($\sigma^2$), but it still represents the same underlying variability.
- Number of Parameters: From its PDF, it is clear that the univariate Gaussian distribution has 2 parameters, namely $\mu$ and $\sigma^2$ (or $\mu$ and $\sigma$).
Parameter Summary
In summary, based on the analysis of their respective probability density functions, we can conclude the following about the number of parameters:
| Distribution | Number of Parameters | Parameters |
|---|---|---|
| Univariate Exponential Distribution | 1 | Rate ($\lambda$) |
| Univariate Gaussian (Normal) Distribution | 2 | Mean ($\mu$), Variance ($\sigma^2$) |
Therefore, the number of parameters in the univariate exponential and Gaussian distributions, respectively, are 1 and 2.
Important Questions from Continuous Distributions
Suppose X is a continuous random variable with probability density function
\(f(x)=\frac{1}{\pi} \frac{1}{1+(x+1)^2}\), -∞ < x < ∞.
Define
\(Y=\left\{\begin{array}{cc} \frac{X}{|X|}, & \text { if } X \neq 0 \\ 0, & \text { if } X=0 \end{array}\right.\)
Then which of the following statements are true?
Let X1, X2, ..., Xn be a random sample from an absolutely continuous distribution with the probability density function
\(f(x \mid \theta)=\left\{\begin{array}{cl} e^{\theta-x}, & \text { if } x \geq \theta \\ 0, & \text { if } x<\theta \end{array},\right.\)
where θ ∈ ℝ is unknown. Define \(\bar{X}=\frac{1}{n} \sum_{i=1}^n X_i\) and X(1) = min{X1, ..., Xn}. Then
which of the following statements are true?
Suppose that X is a continuous random variable with probability density function given by:
f(x) = \(\left\{ {\begin{array}{c} {\frac{x}{8},}&{x \in \left[ {0,2} \right)}\\ {\frac{1}{4},}&{x \in \left[ {2,4} \right)}\\ { - \frac{x}{8} + \frac{3}{4},}&{x \in \left[ {4,6} \right)} \end{array}}\right.\)
Find the mean of X.
The variable x takes a value between 0 and 10 with uniform probability distribution. The variable y takes a value between 0 and 20 with uniform probability distribution. The probability of the sum of variables (x + y) being greater than 20 is _________
Probability density function of a random variable X is given below
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0.25}&{if\;1 \le x \le 5}\\ 0&{otherwise} \end{array}} \right.\)
P (X ≤ 4) is
