Questions: CONTINUOUS RANDOM VARIABLES: PROBABILITY DENSITY FUNCTIONS 1. Weight Gain The graph shows the (subjective) Distribution of x = weight gain probability distribution for the random variable X= "Jack's weight gain over the summer" a) What are the possible values of X ? How do you know? What does that tell you about Jack? b) Which is more likely "Jack gains weight" or "Jack loses weight" ? How do you know? c) Use the graph to make a sensible estimate of Jack's EXPECTED weight gain, E(X). Explain how you found this number.

Solution Steps

The possible values of \( X \) are the values on the x-axis of the graph, which range from -2 to 9. This tells us that Jack's weight gain over the summer can vary from losing 2 pounds to gaining 9 pounds.

To determine whether it is more likely that Jack gains weight or loses weight, we observe the probability distribution. The area under the curve to the right of 0 (positive weight gain) is significantly larger than the area to the left of 0 (weight loss). This indicates that it is more likely for Jack to gain weight than to lose weight.

To estimate Jack's expected weight gain, \( E(X) \), we need to consider the weighted average of all possible values of \( X \), using the probability density function. The graph shows higher probabilities for values around 4 to 6. A reasonable estimate for \( E(X) \) would be around 4 to 5 pounds, as these values have the highest probability density.

Final Answer