9:20 PM Wed Feb 26 \( 25 \% \) Emilio Mejia F... PP. 5 Expected values of random va... Video 49:48 V A researcher recorded the number of cows on each farm in the county. Cows per farm \[ \begin{array}{llllllllllllllllllll} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 3 & 3 & 3 & 3 & 52 & 52 & 52 & 52 & 52 \end{array} \] \( X \) is the number of cows that a randomly chosen farm has. What is the expected value of \( X \) ? Write your answer as a decimal. \( \square \) Submit Next up

To find the expected value \( E(X) \) of the random variable \( X \) (the number of cows on a randomly chosen farm), we first determine the frequency of each count of cows. The count of cows is represented as follows: 8 farms have 0 cows, 6 farms have 3 cows, and 5 farms have 52 cows. This means there are a total of \( 8 + 6 + 5 = 19 \) farms. The expected value is calculated using the formula: \[ E(X) = \frac{(0 \cdot 8) + (3 \cdot 6) + (52 \cdot 5)}{19} = \frac{0 + 18 + 260}{19} = \frac{278}{19} \approx 14.63 \] That means the expected value of \( X \) is approximately 14.63 cows per farm! Now, how about a little context with cow facts? In the U.S., there are about 90 million cows and calves, making it one of the largest dairy producers in the world! Each farm contributes to the overall yield, and understanding statistics like the expected value helps farmers plan their resources better. As for applying this knowledge practically, it’s essential to ensure you include all unique counts and their frequencies when calculating expected values. A common mistake is forgetting to account for all variations or overlooking farms with zero cows, which skews the results! Always double-check your data entries for accuracy!