Math Problem Statement

What does the shaded area represent? P(__ < x < __)

Solution

The graph appears to depict a probability density function (PDF) of a continuous random variable, with a shaded region representing the probability of the variable falling within a specific interval.

Looking at the shaded area on the graph, it seems to extend from x=6x = 6 to x=7x = 7, which means the probability represented by the shaded region is P(6<x<7)P(6 < x < 7).

Thus, the correct answer is:

c. P(6<x<7)P(6 < x < 7)

Would you like more details about how to interpret probability density functions or how this solution was reached?

Here are some questions for further exploration:

  1. How is the total area under a probability density curve interpreted?
  2. What is the difference between cumulative distribution function (CDF) and PDF?
  3. How do we calculate the probability for continuous variables using definite integrals?
  4. What would the graph look like for a different interval, like 7<x<107 < x < 10?
  5. How would this graph change if it represented a discrete probability distribution?

Tip: The area under a probability density function (PDF) over a given interval represents the probability that the random variable falls within that interval.

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Math Problem Analysis

Mathematical Concepts

Probability Density Function (PDF)
Continuous Random Variables
Definite Integration

Formulas

P(a < x < b) = ∫[a,b] f(x) dx

Theorems

Fundamental Theorem of Calculus for continuous probability distributions

Suitable Grade Level

Grades 11-12