Math Problem Statement

Given the following CDF, F(x) = { 0 if x < -1, (x^3 + 1)/2 if -1 <= x <= 1, 1 if x > 1 } Find the following probabilities: a. P(X ≤ 1/2) b. P(-2 < X < 0) c. P(-1/2 < X < 1/2) d. P(X > 1/4) e. P(X = 1/2)

Solution

It looks like you're working with a problem involving a cumulative distribution function (CDF) and need to find certain probabilities. Let's break down the tasks.

The CDF given in the image is:

0 & \text{if } x < -1 \\ \frac{x^3 + 1}{2} & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } x > 1 \end{cases}$$ You've already computed part (a), which involves finding $$P(X \leq \frac{1}{2})$$. Based on the CDF formula, we can calculate it by plugging $$x = \frac{1}{2}$$ into the function. Here's what you've done: $$F\left(\frac{1}{2}\right) = \frac{\left(\frac{1}{2}\right)^3 + 1}{2} = \frac{\frac{1}{8} + 1}{2} = \frac{1.125}{2} = 0.5625$$ So, the correct result for part (a) should be $$P(X \leq \frac{1}{2}) = 0.5625$$, not 0.625. Would you like to go through the other parts of the problem together?

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Math Problem Analysis

Mathematical Concepts

Cumulative Distribution Function (CDF)
Probability
Piecewise Functions

Formulas

F(x) = 0 if x < -1
F(x) = (x^3 + 1)/2 if -1 <= x <= 1
F(x) = 1 if x > 1

Theorems

CDF properties
Probability calculation from CDF

Suitable Grade Level

Grades 11-12 (Advanced High School or Early College)

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