Let X be the random variable with the density function f(x) given by if z20 2.2 f(z) (1+z)" otherwise Find the mean of X. Find the variance of X.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Title: Probability Density Function and Statistical Properties**
**Introduction**
In probability theory, a random variable \( X \) is said to have a probability density function (pdf) \( f(x) \). The pdf provides the probabilities of the possible values of the random variable. Here, we explore the random variable \( X \) with a given pdf and calculate its mean and variance.
### Probability Density Function (pdf) of \( X \)
Let \( X \) be the random variable with the density function \( f(x) \) given by:
\[ f(x) =
\begin{cases}
\frac{2.2}{(1+x)^{1.2}} & \text{if } x \geq 0 \\
0 & \text{otherwise}
\end{cases} \]
In this context, the function is piecewise, meaning \( f(x) \) has different expressions based on the value of \( x \):
- For \( x \geq 0 \), \( f(x) = \frac{2.2}{(1+x)^{1.2}} \)
- For \( x < 0 \), \( f(x) = 0 \)
### Mean and Variance of \( X \)
To further understand \( X \), we are interested in:
1. **The Mean of \( X \)**
2. **The Variance of \( X \)**
These statistical moments provide insights into the expected value and the spread (variability) of the random variable \( X \).
**Steps to Calculate the Mean \( \mu \):**
\[ \mu = E[X] = \int_{-\infty}^{\infty} x \, f(x) \, dx \]
**Steps to Calculate the Variance \( \sigma^2 \):**
\[ \sigma^2 = \text{Var}(X) = E[X^2] - (E[X])^2 \]
These calculations involve integral calculus, making use of the probability density function provided.
**Interactive Exercises:**
- **Find the mean of \( X \):** [Interactive Text Box]
- **Find the variance of \( X \):** [Interactive Text Box]
**Conclusion:**
Understanding the statistical properties (mean and variance) of a random variable with a given density function \( f(x) \) is crucial in probability theory. These properties help in predicting the behavior of the random variable](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb3dca1d-57a3-4839-be8e-2a543baf6bdb%2F48d42b8b-0414-475a-9ba1-1d4a4a637e12%2Filqww_processed.png&w=3840&q=75)
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