Suppose x is a uniform random variable with values ranging from 20 to 70. Find the probability that a randomly selected observation is between 23 and 65.
86
To find the probability that a randomly selected observation is between 23 and 65, we need to calculate the area under the probability density function (PDF) curve between these two values.
Since the variable x is a uniform random variable with values ranging from 20 to 70, the PDF will be a rectangle with a height of 1 divided by the width of the interval (70 - 20).
1) Calculate the width of the interval:
Width = 70 - 20 = 50
2) Calculate the area of the rectangle:
Area = height * width
Area = (1/50) * 50
Area = 1
3) Calculate the area between 23 and 65:
Area_between = (65 - 23) * (1/50)
Area_between = 42 * (1/50)
Area_between = 0.84
So, the probability that a randomly selected observation is between 23 and 65 is 0.84 or 84%.
To find the probability that a randomly selected observation of a uniform random variable, which has a range from 20 to 70, is between 23 and 65, you need to calculate the proportion of the total range that falls within that interval.
The first step is to determine the length of the total range. In this case, the range is from 20 to 70, so the length of the range is 70 - 20 = 50.
Next, determine the length of the interval between 23 and 65. The length of this interval is 65 - 23 = 42.
Finally, calculate the probability by dividing the length of the interval by the length of the total range: P(23 ≤ x ≤ 65) = (42 / 50) = 0.84.
So, the probability that a randomly selected observation falls between 23 and 65 is 0.84 or 84%.