Question

A random variable follows the continuous uniform distribution between 170 and 350. Calculate the following quantities for the distribution.

P(210≤x≤300​) ​decimal places as​ needed.)

Answer 1

To calculate the probability P(210 ≤ x ≤ 300) for a continuous uniform distribution between 170 and 350, we need to find the area under the distribution curve between the values 210 and 300. Using the formula for probability density function of a continuous uniform distribution, we can calculate the probability to be 0.5.

Step-by-step explanation:

The continuous uniform distribution is defined as a probability distribution where every outcome between the minimum and maximum values has an equal chance of occurring. In this case, the random variable follows a continuous uniform distribution between 170 and 350. To calculate the probability P(210 ≤ x ≤ 300), we need to find the area under the distribution curve between the values 210 and 300.

To calculate this probability, we can use the formula:

P(a ≤ x ≤ b) = (∫[a,b] f(x) dx) / (∫[min,max] f(x) dx)

Substituting the given values, we have:

P(210 ≤ x ≤ 300) = ∫[210,300] 1 / (350 - 170) dx

Now, we can evaluate the integral:

P(210 ≤ x ≤ 300) = [x / (350 - 170)] |210300

Substituting the limits of the integral, we get:

P(210 ≤ x ≤ 300) = [(300 / (350 - 170)) - (210 / (350 - 170))]

P(210 ≤ x ≤ 300) = [(300 / 180) - (210 / 180)]

P(210 ≤ x ≤ 300) = 1.6667 - 1.1667

P(210 ≤ x ≤ 300) = 0.5