Question at position 2 To obtain the probability that a continuous random variable falls in a specified range, we find the difference between the cumulative probabilities at the lower and upper end of the:mode.range.median.mean.

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Question at position 2 To obtain the probability that a continuous random variable falls in a specified range, we find the difference between the cumulative probabilities at the lower and upper end of the:mode.range.median.mean.

BlackTom AI · Accepted Answer

The question is about calculating the probability that a continuous random variable falls within a given interval. To obtain this probability, we typically use the cumulative distribution function (CDF), F(x), which gives P(X ≤ x) for any value x.
Option 1: mode. The mode is the value at which the probability density function (PDF) attains its maximum. It does not directly relate to the difference in CDF values to find a range probability, so this is not correct.
Option 2: range. The probability that X falls between a lower bound a and an upper bound b is P(a ≤ X ≤ b) = F(b) − F(a). This exactly uses the difference between the cumulative probabilities at the upper and lower ends of the interval, i.e., the range. This option matches the standard approach.
Option 3: median. The median is the 50th percentile where F(median) = 0.5 for a symmetric distribution, but it does not by itself give the probability of falling within a particular interval. It does not define the difference required for a general range probability.
Option 4: mean. The mean is the average value of the distribution and does not determine the probability content of a specific interval via a simple difference of CDF values. It is not the appropriate endpoint for computing P(a ≤ X ≤ b).
In summary, the correct concept for the probability of landing in a specified range is based on the range endpoints in the CDF, making 'range' the appropriate answer over mode, median, or mean.

Question at position 2 To obtain the probability that a continuous random variable falls in a specified range, we find the difference between the cumulative probabilities at the lower and upper end of the:mode.range.median.mean.

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