Suppose x is a uniform random variable with c=10 and d=60. Find the probability that a randomly selected observation exceeds 30.
What does c and d signify?
To find the probability that a randomly selected observation from a uniform distribution exceeds a certain value, we need to calculate the area of the distribution beyond that value.
Given that x is a uniform random variable with c = 10 and d = 60, we know that the interval of possible values for x is [10, 60].
To calculate the probability of exceeding 30, we first need to find the length of the interval beyond 30. This can be calculated as:
Length of interval beyond 30 = d - 30
Substituting the given values, we have:
Length of interval beyond 30 = 60 - 30 = 30
Since the probability is proportional to the length of the interval, we can calculate the probability of exceeding 30 by dividing the length of the interval beyond 30 by the total length of the interval.
Total length of the interval = d - c
Substituting the given values, we have:
Total length of the interval = 60 - 10 = 50
Therefore, the probability of exceeding 30 is given by:
Probability = Length of interval beyond 30 / Total length of the interval
= 30 / 50
= 0.6
So, the probability that a randomly selected observation exceeds 30 is 0.6, or 60%.
To find the probability that a randomly selected observation exceeds 30, we can use the cumulative distribution function (CDF) of a uniform random variable.
The CDF of a uniform random variable with endpoints c and d is given by:
F(x) = (x - c) / (d - c)
Given c = 10 and d = 60, we can substitute these values into the equation:
F(x) = (x - 10) / (60 - 10)
Now, to find the probability that x exceeds 30, we need to calculate 1 minus the cumulative distribution function when x = 30. Therefore:
P(x > 30) = 1 - F(30)
Substituting x = 30 into the equation for F(x):
P(x > 30) = 1 - [(30 - 10) / (60 - 10)]
Simplify the equation:
P(x > 30) = 1 - (20 / 50)
P(x > 30) = 1 - 0.4
P(x > 30) = 0.6
Hence, the probability that a randomly selected observation from the uniform random variable exceeds 30 is 0.6.