GATE CS 2014 Set 2 Q2 — Probability: Random Variables and Distributions

Let $X$ be the random variable representing the length of the drawn word. The sentence contains $9$ words in total. We list each word and its length below:

• "The" — $3$
• "quick" — $5$
• "brown" — $5$
• "fox" — $3$
• "jumps" — $5$
• "over" — $4$
• "the" — $3$
• "lazy" — $4$
• "dog" — $3$

Since each word is drawn with an equal probability of $\frac{1}{9}$, the expected length $E[X]$ is the average of these lengths:

Rounding to one decimal place, the expected length is 3.9 (or 3.8 if truncated).