Suppose x is a uniform random variable with values ranging from 40 to 60. Find the probability that a randomly selected observation exceeds 54.
To find the probability that a randomly selected observation exceeds 54, we need to calculate the area under the probability density function (PDF) of the uniform distribution from 54 to the maximum value of 60.
The probability density function (PDF) of a uniform random variable is given by:
f(x) = 1 / (b - a)
where 'a' is the minimum value and 'b' is the maximum value of the random variable.
Given that x is a uniform random variable with values ranging from 40 to 60, we have:
a = 40
b = 60
The probability that a randomly selected observation exceeds 54 can be calculated as:
P(X > 54) = âŤ[54, 60] f(x) dx
Since the PDF is constant over the range [40, 60], the integral becomes:
P(X > 54) = âŤ[54, 60] (1 / (60 - 40)) dx
Simplifying,
P(X > 54) = (1 / (60 - 40)) * [x] evaluated from 54 to 60
P(X > 54) = (1 / 20) * [(60 - 54)]
P(X > 54) = (1 / 20) * 6
P(X > 54) = 6 / 20
P(X > 54) = 0.3 or 30%
Therefore, the probability that a randomly selected observation exceeds 54 is 0.3 or 30%.
To find the probability that a randomly selected observation exceeds 54, you need to calculate the proportion of the distribution that is greater than 54.
Since x is a uniform random variable with values ranging from 40 to 60, the probability distribution is evenly spread out over this range.
The total probability of the distribution is 1, since it covers the entire range of possible values.
To calculate the probability of exceeding 54, you need to determine the proportion of the distribution that is greater than 54.
First, you can calculate the width of the distribution by subtracting the lower bound (40) from the upper bound (60).
60 - 40 = 20
Next, you can calculate the width of the portion of the distribution that is greater than 54. This can be done by subtracting 54 from the upper bound and then dividing by the total width.
60 - 54 = 6
6/20 = 0.3
Therefore, the probability that a randomly selected observation exceeds 54 is 0.3, or 30%.
60 - 40 = 20
55 to 60 = 6
Take it from there.