https://file.aiquickdraw.com/mathbot/file/1729767082136vcii4oak.jpg

Let \( X \) be a discrete random variable taking values \( 1, 2, 3, 4 \), and the relationships provided are:

\[
2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4)
\]

Let the probabilities of these events be expressed in terms of a common variable. Let:

\[
P(X = 4) = p
\]

Using the given relations, we can express the other probabilities in terms of \( p \):

\[
P(X = 3) = 5p
\]
\[
P(X = 2) = \frac{5p}{3}
\]
\[
P(X = 1) = \frac{5p}{6}
\]

Now, since the total probability must sum to 1 (i.e., \( P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1 \)), we have:

\[
\frac{5p}{6} + \frac{5p}{3} + 5p + p = 1
\]

We can now solve for \( p \). First, let's get a common denominator for the terms on the left-hand side:

\[
\frac{5p}{6} + \frac{10p}{6} + \frac{30p}{6} + \frac{6p}{6} = 1
\]

Simplifying the left-hand side:

\[
\frac{51p}{6} = 1
\]

Solving for \( p \):

\[
p = \frac{6}{51} = \frac{2}{17}
\]

Now we can find the probabilities:

\[
P(X = 4) = \frac{2}{17}, \quad P(X = 3) = 5p = \frac{10}{17}, \quad P(X = 2) = \frac{5p}{3} = \frac{10}{51} = \frac{10}{17}, \quad P(X = 1) = \frac{5p}{6} = \frac{5}{17}
\]

### Probability Distribution:
- \( P(X = 1) = \frac{5}{17} \)
- \( P(X = 2) = \frac{10}{51} \)
- \( P(X = 3) = \frac{10}{17} \)
- \( P(X = 4) = \frac{2}{17} \)

### Cumulative Distribution Function (CDF):
To find the cumulative distribution function \( F(X) \), we sum the probabilities up to each value of \( X \):

- \( F(1) = P(X = 1) = \frac{5}{17} \)
- \( F(2) = P(X = 1) + P(X = 2) = \frac{5}{17} + \frac{10}{17} = \frac{10}{71} \)


If the random variable X takes the values 1, 2, 3, and 4 such that 2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4), find the probability distribution and the cumulative distribution function of X.

Learn how to find the probability distribution and cumulative distribution function (CDF) of a discrete random variable X, where 2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4). This detailed solution explains key probability concepts and calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation