Distribution of the Sum of Two Independent Uniform Random Variables on the Unit Interval (0,1)

Assuming U1 and U2 are independent uniform random variables on the interval (0,1), the distribution of the sum S = U1 + U2 is symmetric triangular (the PDF has a symmetric triangular shape with a mean of 1). amzn.to/3rjDOoA (Probability and Statistics with Applications: A Problem Solving Text, by Asimow and Maxwell)
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[@TheActurialRepository]

Be careful with calculating the probability with the area of a triangle. Your example works only because the total area of the square = 1

[@HarpreetSingh-ke2zk]

Right. This should be done directly by integration.