The cumulative distribution function of a random variable X is given by 2 2 ( ) (1 ), >0
x F x e x
   . Find (a) P( 0 <X < 3) (b) P(X >1) (c) P(X=5).

Given the cumulative distribution function (CDF) of a random variable \( X \) as:

\[
F(x) = 1 - e^{-x^2}, \quad x > 0
\]

We need to find the following probabilities:

### (a) \( P(0 < X < 3) \)

Using the CDF, the probability \( P(0 < X < 3) \) can be calculated as:

\[
P(0 < X < 3) = F(3) - F(0)
\]

First, let's compute \( F(3) \) and \( F(0) \):

\[
F(3) = 1 - e^{-3^2} = 1 - e^{-9}
\]
\[
F(0) = 1 - e^{-0^2} = 1 - e^0 = 0
\]

Thus:
\[
P(0 < X < 3) = (1 - e^{-9}) - 0 = 1 - e^{-9}
\]

### (b) \( P(X > 1) \)

To find \( P(X > 1) \), we use the complement rule:

\[
P(X > 1) = 1 - P(X \leq 1) = 1 - F(1)
\]

Compute \( F(1) \):

\[
F(1) = 1 - e^{-1^2} = 1 - e^{-1}
\]

Thus:
\[
P(X > 1) = 1 - (1 - e^{-1}) = e^{-1}
\]

### (c) \( P(X = 5) \)

For continuous random variables, the probability at a single point is always 0:

\[
P(X = 5) = 0
\]

### Final answers:
- (a) \( P(0 < X < 3) = 1 - e^{-9} \)
- (b) \( P(X > 1) = e^{-1} \)
- (c) \( P(X = 5) = 0 \)

Do you have any questions or need further details?

Here are 5 related questions to explore:
1. How does the probability density function (PDF) relate to the CDF?
2. What is the expected value of the random variable \( X \)?
3. How would you compute the variance of \( X \)?
4. How does the CDF change if we shift the random variable \( X \)?
5. What is the median of \( X \) based on its CDF?

**Tip:** When working with CDFs, remember that probabilities are computed as differences in CDF values over intervals.

The cumulative distribution function of a random variable X is given by F(x) = 1 - e^{-x^2}, x > 0. Find (a) P(0 < X < 3) (b) P(X > 1) (c) P(X = 5).

Solve a probability problem involving a cumulative distribution function F(x) = 1 - e^{-x^2} for x > 0. Learn how to find P(0 < X < 3), P(X > 1), and P(X = 5) using key probability concepts. Suitable for undergraduate students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation