Some Questions Based on Continuous Probability Distributions
Question
Compute the conditional probability:
\[ P\left(X > \frac{3}{4} \mid X > \frac{1}{2}\right) \]
Theory Behind Solution
The conditional probability formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
For continuous random variables, probability is computed using integration.
Understanding Probability Density Functions
A probability density function (p.d.f.) must satisfy:
\[ \int_{-\infty}^{\infty} f(x) \,dx = 1 \]
The area under the p.d.f. curve represents probabilities.
Solution – Part 1
Compute:
\[ P\left(X > \frac{3}{4} \mid X > \frac{1}{2}\right) = \frac{P\left(X > \frac{3}{4}\right)}{P\left(X > \frac{1}{2}\right)} \]
Given probability density function (p.d.f.):
\[ f(x) = 4x^3, \quad 0 \leq x \leq 1 \]
Solution – Part 2
Solving for \( P(X > \frac{3}{4}) \):
\[ \left[ x^4 \right]_{3/4}^{1} = 1 – \left(\frac{3}{4}\right)^4 = \frac{175}{256} \]
Solving for \( P(X > \frac{1}{2}) \):
\[ \left[ x^4 \right]_{1/2}^{1} = 1 – \left(\frac{1}{2}\right)^4 = \frac{15}{16} \]
Compute final result:
\[ \frac{\frac{175}{256}}{\frac{15}{16}} = \frac{35}{48} \]
Example: P.D.F. of Ghee Pack Weights
The probability density function (p.d.f.) of a “1-litre pure ghee pack” is given by:
\[ f(x) = \begin{cases} 200(x – 1), & 1 \leq x \leq 1.1 \\ 0, & \text{otherwise} \end{cases} \]
Examine whether the given p.d.f. is valid and compute \( P[1.01 < X < 1.02] \).
Solution to Example
Compute:
\[ \int_{1}^{1.1} 200(x – 1) \,dx \]
Solving:
\[ \left[ 100(x – 1)^2 \right]_{1}^{1.1} = 100(1.1 – 1)^2 – 100(1 – 1)^2 = 1 \]
Since the integral evaluates to 1, the given function is a valid p.d.f.
Compute \( P[1.01 < X < 1.02] \):
\[ \int_{1.01}^{1.02} 200(x – 1) \,dx \]
Evaluating:
\[ 100(1.02 – 1)^2 – 100(1.01 – 1)^2 = 0.03 \]
Key Takeaways
- Conditional probability helps refine probability calculations given additional information.
- Integration is essential in computing probabilities for continuous random variables.
- Verifying a p.d.f. involves checking that it integrates to 1.
- Practical applications include real-world weight distributions and reliability analysis.
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