a. Express P (x)= 1/6 ; for x = 1, 2, 3, 4, 5, 6, in distribution form.
b. Construct a histogram of the probability distribution P(x) =1 over 6 ; for x = 1, 2, 3, 4, 5, 6.
c. Describe the shape of the histogram in part (b).
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a. To express the probability distribution P(x) = 1/6 for x = 1, 2, 3, 4, 5, 6 in distribution form, we can represent it using the following table:
x | P(x)
---------
1 | 1/6
2 | 1/6
3 | 1/6
4 | 1/6
5 | 1/6
6 | 1/6
b. To construct a histogram of the probability distribution P(x) = 1/6 for x = 1, 2, 3, 4, 5, 6, we can use the following steps:
1. Draw a vertical axis representing the probability density.
2. Draw a horizontal axis representing the possible outcomes (x values).
3. Create bars to represent each outcome (x) using the probability density as the height of the bars.
The histogram will look as follows:
|
| **
| **
| **
| **
| **
| **
|_____________________
1 2 3 4 5 6
Each asterisk (*) represents a bar with the same height, indicating that the probability of each outcome (x) is 1/6.
c. The shape of the histogram constructed in part (b) is rectangular, where each bar has the same height. This shape indicates that the probability distribution is equally likely for each outcome (x = 1, 2, 3, 4, 5, 6).
a. To express P(x) = 1/6 in distribution form for x = 1, 2, 3, 4, 5, 6, we can write it as:
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
b. To construct a histogram of the probability distribution P(x) = 1/6 for x = 1, 2, 3, 4, 5, 6, we need to plot the values of x on the x-axis and their corresponding probabilities on the y-axis.
First, draw a horizontal line (x-axis) and a vertical line (y-axis) to create a coordinate system.
On the x-axis, label the numbers 1, 2, 3, 4, 5, 6.
On the y-axis, label the range of probabilities from 0 to 1.
Then, for each value of x, draw a rectangle above that number on the x-axis with a height equal to the corresponding probability.
For example, for P(1) = 1/6, draw a rectangle above the number 1 on the x-axis with a height of 1/6.
Repeat this process for all values of x: 2, 3, 4, 5, and 6.
c. The shape of the histogram in part (b) would be a uniform distribution. This means that all values of x (1, 2, 3, 4, 5, 6) have the same probability of 1/6, resulting in equal heights for each rectangle in the histogram. This shape indicates that the probability is evenly distributed among the possible outcomes.