Question

finding the mean of a probability distribution
x | probability: p(x)
0 | 0.1
1 | 0.2
2 | 0.4
3 | 0.2
4 | 0.1
using the data from the table, what is p(3)?
what is the mean of the probability distribution?

Explanation:

Part 1: Find \( P(X = 3) \)

Step 1: Locate \( X = 3 \) in the table

We look at the row where \( X = 3 \) and find the corresponding probability \( P(X) \). From the table, when \( X = 3 \), \( P(X) = 0.2 \).

The formula for the mean \( \mu \) of a discrete probability distribution is \( \mu = \sum [X \cdot P(X)] \), where we multiply each value of \( X \) by its corresponding probability \( P(X) \) and then sum the products.

Step 1: Calculate \( X \cdot P(X) \) for each row

• For \( X = 0 \): \( 0 \times 0.1 = 0 \)
• For \( X = 1 \): \( 1 \times 0.2 = 0.2 \)
• For \( X = 2 \): \( 2 \times 0.4 = 0.8 \)
• For \( X = 3 \): \( 3 \times 0.2 = 0.6 \)
• For \( X = 4 \): \( 4 \times 0.1 = 0.4 \)

Step 2: Sum the products

Now we sum these results: \( 0 + 0.2 + 0.8 + 0.6 + 0.4 \)
First, \( 0 + 0.2 = 0.2 \)
Then, \( 0.2 + 0.8 = 1.0 \)
Next, \( 1.0 + 0.6 = 1.6 \)
Finally, \( 1.6 + 0.4 = 2.0 \)

Answer:

\( P(X = 3) = 0.2 \)

Part 2: Find the mean of the probability distribution