A continuous uniform distribution is a continuous distribution where the probability of all outcomes is equal. It plays an important role in Monte Carlo simulation.
The following formula gives the pdf for a continuous uniform distribution ranging from a to b:
The following formula gives the cumulative distribution formula (cdf):
F(x) = 0 for x ≤ a
F(x) = (x-a)/(b-a) for a< x< b
F(x) = 1 for x≥b
The following formulas give the cumulative probability that the value lies between x1 and x2:
P(x1 ≤ X ≤x2) = (x2-x1)/(b-a) if a ≤ x1 < x2 ≤ b
P(x1 ≤ X ≤x2) = (x2-a)/(b-a) if x1 ≤ a < x2 ≤ b
P(x1 ≤ X ≤x2) = (b-x1)/(b-a) if a ≤ x1 < b ≤ x2
P(x1 ≤ X ≤x2) = 1 if x1 ≤ a < b ≤ x2
P(x1 ≤ X ≤x2) = 0 otherwise
Do not get confused with the formulas. You need not remember the formulas if you can understand the concept behind that formulas. The continuous probability distribution area is like a rectangle with (b-a) as its one side and 1/(b-a) as its other side.
Chart:
While calculating the probability, we just need to calculate how much of the area is covered by the given number. When the number x1 and x2 are between a and b, then those numbers are going to cover only x2 - x1 of the width as the height will always remain the same at 1/(b-a). So, the total area covered will be (x2 - x1)/(b-a).
Similarly, if x1 is greater than a and less than b, but x2 is greater than b, then the width covered will be equal to b - x1 because the continuous random variable does not like beyond b and the area covered will be equal to (b - x1)/(b-a).
The mean of the continuous uniformly distributed variable = (a+b)/2
Variance of the continuous uniformly distributed variable = (b-a)2/12
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Example 4: Calculating probabilities for a continuous uniform distribution
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A fixed income portfolio manager has taken a position in the below-investment grade bonds. The payoff of the bonds can range from -$100 million to $200 million. The payoff is assumed to be continuous uniformly distributed.
Compute the expected value of the payoff and the probability that the payoff will be:
(a) Positive
(b) Negative
(c) Greater than $100 million
(d) Between -$50 million and $25 million
Solution:
The expected value of the payoff = (-100 + 200)/2 = $50 million
(a) Probability that the payoff is positive, P(X>0) = (200-0)/[200 - (-100)] = 200/300 = 0.667.
(b) Probability that the payoff is negative, P(X<0) = [0 - (-100)]/[200 - (-100)] = 100/300 = 0.333.
(c) Probability that the payoff is greater than $100 million, P(X>100) = (200-100)/[200 - (-100)] = 100/300 = 0.333.
(d) Probability that the payoff is between -$50 million and $25 million, P(-50< X <25) = [25 - (-50)]/[200 - (-100)] = 75/300 = 0.25.
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