Given the probability density function f(x) = value, the mean, the variance and the standard deviation. a² over the interval [3, 8], find the expected 485 Expected value: Mean: Variance: Standard Deviation:
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!
![**Probability Density Function Problem**
Given the probability density function \( f(x) = \frac{3}{485} x^2 \) over the interval \([3, 8]\), find the expected value, the mean, the variance, and the standard deviation.
1. **Expected Value:**
\[ \text{Expected value:} \quad \_\_\_\_\_\_\_\_\_\_ \]
2. **Mean:**
\[ \text{Mean:} \quad \_\_\_\_\_\_\_\_\_\_ \]
3. **Variance:**
\[ \text{Variance:} \quad \_\_\_\_\_\_\_\_\_\_ \]
4. **Standard Deviation:**
\[ \text{Standard Deviation:} \quad \_\_\_\_\_\_\_\_\_\_ \]
**Note:** This problem involves a probability density function (PDF), which is typically used to describe the likelihood of different outcomes in a continuous random variable. The calculations for the expected value, mean, variance, and standard deviation will involve integrating the function \(f(x)\) over the given interval \([3, 8]\).
**Graph Explanation:**
There are no graphs or diagrams included in the provided material. However, if there were, an explanation would involve detailing the axes, labels, and the shape or trend represented by the graph to facilitate understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb3dca1d-57a3-4839-be8e-2a543baf6bdb%2F67c4c539-079a-40cc-a2a9-3029d93a5636%2F2hjo1s_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
