Discrete Uniform Distribution

• Discrete Uniform Distribution: If the random variable $X$ assumes the values $x_1,x_2, \ldots x_n$, with equal probabilities, then the discrete uniform distribution (probability mass function) is given by
  $$f(x;k)=\frac{1}{k},~x=x_1,x_2, \ldots x_k$$

• Example 5.1: When a light bulb is selected at random from a box that contains a 40-watt bulb, a 60-watt bulb, a 75-watt bulb, and a 100-watt bulb, each element of the sample space $S = \{40, 60, 75, 100\}$ occurs with probability $1/4$.
• Therefore, we have a uniform distribution, with
  $$f(x;4)=\frac{1}{4},~x=40,60,75,100$$

• Example 5.2: When a die is tossed, each element of the sample space $S = \{1, 2, 3, 4, 5, 6\}$ occurs with probability $1/6$.

• Example 5.2:Cont.
  • Therefore, we have a uniform distribution, with
    $$f(x;6)=\frac{1}{6},~x=1,2,3,4,5,6$$

  Figure 1: Histogram for the tossing of a die.

  

• Theorem 5.1::
  
• Example 5.3: Referring to Example 5.2 (tossing a die), we find that
  $$\mu=3.5,~~\sigma^2=2.92$$