Math Problem Statement
Solution
To solve this problem, let's go through each part systematically.
Given Data:
The probability density function (PDF) of a continuous random variable is provided as:
Objective:
- Find the cumulative distribution function (CDF) .
- Calculate the probabilities:
- (a)
- (b)
- (c)
- (d)
- (e) Find such that .
Step-by-Step Solution
Step 1: Find the Cumulative Distribution Function (CDF)
To find , the cumulative distribution function, we integrate over the range to :
-
Integrate : Calculate .
-
Evaluate the definite integral from to : Substitute the limits of integration.
Step 2: Calculate the Probabilities
Once we have the CDF , we can use it to calculate each probability as follows:
- (a) : Calculate .
- (b) : Since the maximum of is 8, .
- (c) : Calculate .
- (d) : Calculate .
- (e) Find such that : Solve for in the equation .
I’ll perform the calculations for these integrals and probabilities now.Here are the results for each part:
-
CDF :
-
Calculated Probabilities:
- (a)
- (b) (since the maximum range of is 8, so )
- (c)
- (d)
-
(e) To find such that :
- The solution to yields .
Let me know if you would like a detailed explanation of any step, or if you have further questions!
Here are some related questions for deeper understanding:
- How is the cumulative distribution function derived from a probability density function?
- What does it mean when in this context?
- How can you interpret the probability in terms of the distribution?
- Why is the solution for when significant in probability?
- What are some applications of calculating specific probabilities for continuous distributions?
Tip: When integrating probability density functions, always check that the function integrates to 1 over the defined interval, confirming it as a valid PDF.
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Math Problem Analysis
Mathematical Concepts
Probability
Continuous Probability Distributions
Cumulative Distribution Function (CDF)
Integration
Formulas
f(x) = 3(8x - x^2) / 256
F(x) = ∫f(t) dt from 0 to x
P(a < X < b) = F(b) - F(a)
P(X > x) = 1 - F(x)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
University Level
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