For a continuous random variable X, P(23 ≤ X ≤ 66) = 0.25 and P(X > 66) = 0.14. Calculate the following probabilities. (Round your answers to 2 decimal places.)
a) P(X<66)
b) P(X<23)
c) P(X=66)
a) P(X<66) = 1 - P(X>66)
= 1 - 0.14
= 0.86
b) P(X<23) = 1 - P(23 ≤ X ≤ 66)
= 1 - 0.25
= 0.75
c) P(X=66) = P(X<66) - P(X<23)
= 0.86 - 0.75
= 0.11
To calculate the requested probabilities, we'll use the properties of the cumulative distribution function (CDF) of a continuous random variable.
a) P(X<66):
Since we know that P(X>66) = 0.14, we can subtract this value from 1 to find the probability of X being less than 66:
P(X<66) = 1 - P(X>66)
P(X<66) = 1 - 0.14
P(X<66) = 0.86
b) P(X<23):
Given that P(23 ≤ X ≤ 66) = 0.25, we can subtract this probability from 1 to find the probability of X being less than 23:
P(X<23) = 1 - P(23 ≤ X ≤ 66)
P(X<23) = 1 - 0.25
P(X<23) = 0.75
c) P(X=66):
The probability of X taking on a specific value for a continuous random variable is always zero. Hence, P(X=66) = 0.
To calculate the probabilities, we need to use the properties of probability distributions for continuous random variables.
a) P(X < 66): Since X is a continuous random variable, P(X < 66) can be found by subtracting the probability of X being greater than 66 from 1. So,
P(X < 66) = 1 - P(X > 66)
Given that P(X > 66) = 0.14, we can substitute this value into the equation:
P(X < 66) = 1 - 0.14 = 0.86
Therefore, P(X < 66) is equal to 0.86.
b) P(X < 23): Since 23 is the lower bound in the interval given in the question, P(X < 23) will be zero because the range of X is not defined for values less than the lower bound.
Therefore, P(X < 23) is equal to 0.
c) P(X = 66): For a continuous random variable, the probability of a precise value (in this case, 66) is zero. This is because the range of a continuous random variable extends to an infinite number of values, making the probability of any single precise value being obtained very low.
Therefore, P(X = 66) is equal to 0.