The Barron's Big Money Poll asked 131 investment managers across the United States about their short-term investment outlook (Barron's, October 28, 2002). Their responses showed 4% were very bullish, 39% were bullish, 29% were neutral, 21% were bearish, and 7% were very bearish. Let x be the random variable reflecting the level of optimism about the market. Set x = 5 for very bullish down through x = 1 for very bearish.

Develop a probability distribution for the level of optimism of investment managers.

Compute the expected value for the level of optimism.

Compute the variance and standard deviation for the level of optimism.

Comment on what your results imply about the level of optimism and its variability.