Uniform Distributions

Theory

Uniform distribution

A random variable is called a uniform random variable the interval (a,b) if its probability density function is given

\[f_X(x) = \begin{cases} {1 \over b-a } \\ 0 \end{cases} \]

Worked Examples

Worked Examples

The weights to salads taken by customers at a self-service salad bare found to be uniformly distributed between 250g and 450g

Exercises
  1. Write down the probability density function for this distribution.
  2. Find the probability that a customer will take a salad weighing more than 400g.
  3. Find the probability that a customer will take a salad weighing between 320g and 400g.
  4. What are the mean and standard deviations of the salad weights taken by customers.

Distributional Formulas

Continuous Uniform distribution

The probability density function is given as

\[f(x) = {1 \over b-a} \mbox{ for } a \le x \le b\]

For any value “c” between the minimum value a and the maximum value b

\[P(X \ geq c) = {b-c \over b-a}\]

here b is the upper bound while c is the lower bound

\[P(X \ leq c) = {c-a \over b-a}\]

here c is the upper bound while a is the lower bound

Probability of an outcome being between lower bound L and upper bound U \[P( L \leq X \leq U) = { U - L \over b – a }\]

Reminder

  • [L] lower bound of an interval

  • [U] upper bound of an interval

  • ” \(\leq\)” is less than or equal to

  • ” \(\geq\)” is greater than or equal to

  • \(L \leq X \leq U\) simply states that X is between L and U inclusively.

(“inclusively” mean that X could be exactly L or U also, although the probability of this is extremely low)