a. Since X follows a uniform distribution, the mean is equal to the average of the lower and upper limits.
Mean = (Lower limit + Upper limit) / 2 = (730 + 830) / 2 = 780
The standard deviation for a uniform distribution is given by the formula:
Standard deviation = (Upper limit - Lower limit) / sqrt(12) = (830 - 730) / sqrt(12) ≈ 10.240
Therefore, the mean is 780 and the standard deviation is approximately 10.240.
b. To find the probability that X is less than 820, we need to find the area under the probability density function (PDF) curve from the lower limit (730) to 820.
The PDF for a uniform distribution is equal to 1 / (Upper limit - Lower limit) in the range between the lower and upper limits.
Therefore, the probability that X is less than 820 can be calculated as follows:
Probability(X < 820) = (820 - 730) / (830 - 730) = 90 / 100 = 0.9
So, the probability that X is less than 820 is 0.9.
A random variable X follows the uniform distribution with a lower limit of 730 and an upper limit of 830.
a. Calculate the mean and the standard deviation for the distribution. (Round intermediate calculation for standard deviation to 4 decimal places and final answer to 2 decimal places.)
b. What is the probability that X is less than 820? (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
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