Question

the probability distribution below is

Explanation:

Since the problem is about a probability distribution (likely to find the expected value or verify the sum of probabilities, etc.), but the exact question is not fully clear. Assuming we need to check if it's a valid probability distribution (sum of \( P_X(x) = 1 \)) or find the expected value \( E(X)=\sum x\cdot P_X(x) \). Let's assume we need to find the expected value. From the graph:

• For \( x = 1 \), \( P_X(1) \approx 0.05 \) (since the bar is at 0.05)
• For \( x = 2 \), \( P_X(2) \approx 0.1 \)
• For \( x = 3 \), \( P_X(3) \approx 0.15 \)
• For \( x = 4 \), \( P_X(4) \approx 0.2 \)
• For \( x = 5 \), \( P_X(5) \approx 0.25 \)
• For \( x = 6 \), \( P_X(6) \approx 0.35 \) (Wait, but let's check the sum: \( 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.35 = 1.1 \), which is more than 1. Maybe the heights are: \( x=1 \): ~0.05, \( x=2 \): ~0.1, \( x=3 \): ~0.15, \( x=4 \): ~0.2, \( x=5 \): ~0.25, \( x=6 \): ~0.3? Wait, the y-axis has 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4. Let's re - estimate:

Looking at the bars:

• \( x = 1 \): bar height ~0.05 (so \( P(1)=0.05 \))
• \( x = 2 \): bar height ~0.1 (so \( P(2)=0.1 \))
• \( x = 3 \): bar height ~0.15 (so \( P(3)=0.15 \))
• \( x = 4 \): bar height ~0.2 (so \( P(4)=0.2 \))
• \( x = 5 \): bar height ~0.25 (so \( P(5)=0.25 \))
• \( x = 6 \): bar height ~0.35? Wait, no, the sum should be 1. Let's recalculate: \( 0.05+0.1 + 0.15+0.2+0.25 + 0.25=1 \). Maybe the bar for \( x = 6 \) is 0.25? Wait, the image shows the bar for \( x = 6 \) reaching up to 0.35? Maybe the question is to find the expected value. Let's proceed with the given heights:

Step 1: Identify \( x \) and \( P_X(x) \)

\( x \)	1	2	3	4	5	6

Assuming the correct probabilities (maybe the bar for \( x = 6 \) is 0.25):

Step 2: Calculate \( E(X)=\sum_{i = 1}^{n}x_i\cdot P_X(x_i) \)

\( E(X)=1\times0.05 + 2\times0.1+3\times0.15 + 4\times0.2+5\times0.25+6\times0.25 \)

Step 3: Compute each term

• \( 1\times0.05 = 0.05 \)
• \( 2\times0.1=0.2 \)
• \( 3\times0.15 = 0.45 \)
• \( 4\times0.2 = 0.8 \)
• \( 5\times0.25=1.25 \)
• \( 6\times0.25 = 1.5 \)

Step 4: Sum the terms

\( E(X)=0.05 + 0.2+0.45 + 0.8+1.25+1.5=4.25 \)

If we use the original (incorrect sum) probabilities:

\( E(X)=1\times0.05 + 2\times0.1+3\times0.15 + 4\times0.2+5\times0.25+6\times0.35 \)

\( E(X)=0.05+0.2 + 0.45+0.8+1.25 + 2.1=4.85 \)

Since the problem statement is not fully clear, but if we assume the task is to check the sum of probabilities (to see if it's a valid PMF), the sum with the given bar heights (as per the graph) is \( 0.05 + 0.1+0.15 + 0.2+0.25+0.35 = 1.1 \), which is not equal to 1, so it's not a valid probability distribution. If we assume a correction (e.g., \( P(6)=0.25 \)), the sum is 1.

If the question was to find the expected value, and we take the corrected probabilities (sum to 1), the expected value is 4.25. If we take the given (incorrect sum) probabilities, it's 4.85.

Since the problem is not fully specified, but based on the graph (probability distribution of a discrete random variable), the key steps would involve identifying the probabilities for each \( x \) and then performing the relevant calculation (sum of probabilities or expected value).

If we assume the question is "Is this a valid probability distribution?"

Step1: Identify probabilities

For \( x = 1,2,3,4,5,6 \), \( P(x) \) are ~0.05, 0.1, 0.15, 0.2, 0.25, 0.35.

Step2: Sum probabilities

\( 0.05 + 0.1+0.15 + 0.2+0.25+0.35 = 1.1 \)

Step3: Check validity

A valid PMF has \( \sum P(x)=1 \). Here, sum = 1.1 ≠ 1, so not valid.

Answer:

No (if we use the given bar heights as probabilities, sum = 1.1 ≠ 1)