The​ random-number generator on calculators randomly generates a number between 0 and 1. The random variable​ X, the number​ generated, follows a uniform probability distribution.
​(a) Identify the graph of the uniform density function.
​(b) What is the probability of generating a number between 0.53 and 0.95​?
​(c) What is the probability of generating a number greater than 0.83​?

This problem describes a uniform probability distribution for a random variable \( X \), which represents the output of a random-number generator that generates numbers between 0 and 1.

### (a) Identify the graph of the uniform density function.
Since the random variable \( X \) follows a **uniform probability distribution** between 0 and 1, the probability density function (PDF) for this uniform distribution is constant over the interval [0, 1].

The uniform distribution's PDF for \( X \) is given by:

\[
f(x) = 
\begin{cases}
1, & 0 \leq x \leq 1 \\
0, & \text{otherwise}
\end{cases}
\]

Graphically, this is a horizontal line at height 1 from \( x = 0 \) to \( x = 1 \), forming a rectangle over the interval [0, 1].

### (b) What is the probability of generating a number between 0.53 and 0.95?
In a uniform distribution, the probability is simply the length of the interval over the total range. So the probability of generating a number between 0.53 and 0.95 is the length of this interval:

\[
P(0.53 \leq X \leq 0.95) = \text{length of the interval} = 0.95 - 0.53 = 0.42
\]

Thus, the probability of generating a number between 0.53 and 0.95 is 0.42.

### (c) What is the probability of generating a number greater than 0.83?
Similarly, the probability of generating a number greater than 0.83 is the length of the interval from 0.83 to 1:

\[
P(X > 0.83) = 1 - 0.83 = 0.17
\]

Thus, the probability of generating a number greater than 0.83 is 0.17.

---

Do you need any further explanation or details?

Here are 5 related questions:
1. What is the probability of generating a number less than 0.25?
2. How do we calculate the expected value for a uniform distribution?
3. What is the cumulative distribution function (CDF) for a uniform distribution?
4. How does the shape of the density function change if the uniform distribution is on an interval [a, b] instead of [0, 1]?
5. What is the variance of the uniform distribution on [0, 1]?

**Tip**: In a uniform distribution, probabilities are simply the relative lengths of intervals within the total range.

The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution. (a) Identify the graph of the uniform density function. (b) What is the probability of generating a number between 0.53 and 0.95? (c) What is the probability of generating a number greater than 0.83?

Learn how to calculate probabilities for a uniform distribution with a random number generator. This problem covers identifying the uniform density function and calculating the probability for specific intervals. Suitable for Grades 10-12, this guide explains key concepts in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation