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

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

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Write equations of cosine functions using properties

Full solution

Q.

1

.) Let

X

be a uniform discrete random variable which takes on the possibl values of

a, a+1, a+2, \ldots, b

. Calculate

E(X)

and

\operatorname{Var}(X)

.

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

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