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Let X be a uniform discrete random variable which takes on the possible values of a,a+1,a+2,dots,b. Calculate E(X) and Var(X).

Let X X be a uniform discrete random variable which takes on the possible values of a,a+1,a+2,…,b a, a+1, a+2, \ldots, b . Calculate E(X) E(X) and Var⁡(X) \operatorname{Var}(X) .

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Q. Let X X be a uniform discrete random variable which takes on the possible values of a,a+1,a+2,…,b a, a+1, a+2, \ldots, b . Calculate E(X) E(X) and Var⁡(X) \operatorname{Var}(X) .
  1. Define XX values: Define the random variable XX and its possible values.XX is a uniform discrete random variable which can take on the values aa, a+1a+1, a+2a+2, ..., bb. This means that each value is equally likely to occur.
  2. Calculate E(X): Calculate the expected value E(X)E(X). The expected value of a uniform discrete random variable is the average of all possible values. To find E(X)E(X), we sum all the values from aa to bb and divide by the number of values.\newlineE(X)=a+(a+1)+(a+2)+…+bb−a+1E(X) = \frac{a + (a+1) + (a+2) + \ldots + b}{b - a + 1}\newlineThis is an arithmetic series with the first term aa, the last term bb, and the number of terms b−a+1b - a + 1.\newlineE(X)=[a+b2]∗b−a+1b−a+1E(X) = \left[\frac{a + b}{2}\right] * \frac{b - a + 1}{b - a + 1}\newlineE(X)=a+b2E(X) = \frac{a + b}{2}
  3. Calculate Var(X): Calculate the variance Var(X)Var(X). The variance of a uniform discrete random variable is given by the formula: Var(X)=[(b−a+1)2−1]12Var(X) = \frac{[(b - a + 1)^2 - 1]}{12} This formula comes from the fact that the variance is the average of the squared differences from the mean for all possible values. Var(X)=[(b−a+1)2−1]12Var(X) = \frac{[(b - a + 1)^2 - 1]}{12}

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