Suppose x is a uniform random variable with c=10 and d=70. Find the probability that a randomly selected observation is between 13 and 65.
a. 0.133
b. 0.867
c. 0.8
d. 0.5
To find the probability that a randomly selected observation is between 13 and 65, you need to calculate the cumulative probability at the upper value (65) and subtract the cumulative probability at the lower value (13).
The formula to calculate the cumulative probability for a continuous uniform distribution is:
P(x ≤ a) = (a - c) / (d - c)
Where:
P(x ≤ a) - Cumulative probability at a given value (a)
a - Upper value
c - Lower value
d - Upper limit of the distribution
Note that in a continuous uniform distribution, the probability is evenly spread across the interval between the lower and upper values.
Now let's substitute the given values into the formula:
P(x ≤ 65) = (65 - 10) / (70 - 10) = 55 / 60 = 0.917
P(x ≤ 13) = (13 - 10) / (70 - 10) = 3 / 60 = 0.05
To find the probability between 13 and 65, subtract the cumulative probability at 13 from the cumulative probability at 65:
P(13 < x ≤ 65) = P(x ≤ 65) - P(x ≤ 13) = 0.917 - 0.05 = 0.867
So the correct option is b. 0.867