To find the median of a probability distribution, we need to determine the value that corresponds to the 50th percentile, i.e., the value at which the cumulative probability reaches or exceeds 0.5.
Let us carefully calculate the cumulative probabilities for each grade.
Given the grades and their respective probabilities:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline \text{Grade} & 4 & 3 & 2 & 1 & 0 \\
\hline \text{Probability} & 0.4 & 0.32 & 0.17 & 0.08 & 0.03 \\
\hline
\end{array}
\][/tex]
1. Compute the cumulative probabilities:
- The cumulative probability for a grade of 4: [tex]\( P(X \leq 4) = 0.4 \)[/tex]
- The cumulative probability for a grade of 3: [tex]\( P(X \leq 3) = 0.4 + 0.32 = 0.72 \)[/tex]
- The cumulative probability for a grade of 2: [tex]\( P(X \leq 2) = 0.72 + 0.17 = 0.89 \)[/tex]
- The cumulative probability for a grade of 1: [tex]\( P(X \leq 1) = 0.89 + 0.08 = 0.97 \)[/tex]
- The cumulative probability for a grade of 0: [tex]\( P(X \leq 0) = 0.97 + 0.03 = 1.0 \)[/tex]
We can summarize these cumulative probabilities in the following cumulative distribution table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline \text{Grade} & 4 & 3 & 2 & 1 & 0 \\
\hline \text{Cumulative Probability} & 0.4 & 0.72 & 0.89 & 0.97 & 1.0 \\
\hline
\end{array}
\][/tex]
2. Determine the median of the distribution:
- The median is the value at which the cumulative probability first reaches or exceeds 0.5.
Looking at the cumulative probabilities:
- For Grade 4, the cumulative probability is 0.4, which is less than 0.5.
- For Grade 3, the cumulative probability is 0.72, which is greater than 0.5.
Therefore, the smallest grade with a cumulative probability that is at least 0.5 is 3.
Conclusion:
The median of the distribution is 3.
