Question 1177291: A random variable X has the following distribution:
x = −5 0 3 6
p(x) = 0.2 0.1 0.4 0.3
Find the cumulative distribution function F (x) and graph it.
Thank You
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! The cumulative distribution function (CDF), denoted as F(x), is the probability that a random variable X takes on a value less than or equal to x. Here's how to find the CDF for the given distribution and graph it:
**1. Calculate the CDF**
* **F(-5) = P(X ≤ -5) = 0.2** (The probability of X being -5)
* **F(0) = P(X ≤ 0) = 0.2 + 0.1 = 0.3** (The probability of X being -5 or 0)
* **F(3) = P(X ≤ 3) = 0.2 + 0.1 + 0.4 = 0.7** (The probability of X being -5, 0, or 3)
* **F(6) = P(X ≤ 6) = 0.2 + 0.1 + 0.4 + 0.3 = 1** (The probability of X being -5, 0, 3, or 6)
**2. Define the CDF Function**
The CDF, F(x), is a piecewise function defined as follows:
* F(x) = 0, for x < -5
* F(x) = 0.2, for -5 ≤ x < 0
* F(x) = 0.3, for 0 ≤ x < 3
* F(x) = 0.7, for 3 ≤ x < 6
* F(x) = 1, for x ≥ 6
**3. Graph the CDF**
To graph the CDF:
* Plot the points (-5, 0.2), (0, 0.3), (3, 0.7), and (6, 1).
* Draw a horizontal line segment from (-∞, 0) to (-5, 0).
* Draw a horizontal line segment from (-5, 0.2) to (0, 0.2).
* Draw a horizontal line segment from (0, 0.3) to (3, 0.3).
* Draw a horizontal line segment from (3, 0.7) to (6, 0.7).
* Draw a horizontal line segment from (6, 1) to (∞, 1).
**The graph will look like a staircase with jumps at the values -5, 0, 3, and 6. The height of each jump corresponds to the probability of X taking that value.**
Let me know if you would like a visual representation of the graph. I can provide instructions on how to create it using a tool like Google Sheets or other graphing software.
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