Let X be a random variable that takes on values between 0...

Let $X$ be a random variable that takes on values between 0 and $c$. That is, $P\{0 \leq X \leq c\}=1$. Show that $$ \operatorname{Var}(X) \leq \frac{c^{2}}{4} $$ HINT: One approach is to first argue that $$ E\left[X^{2}\right] \leq c E[X] $$ Then use this to show that $$ \operatorname{Var}(X) \leq c^{2}[\alpha(1-\alpha)] \quad \text { where } \quad \alpha=\frac{E[X]}{c} $$

Let $X$ be a random variable that takes on values between 0 and $c$. That is, $P\{0 \leq X \leq c\}=1$. Show that
$$
\operatorname{Var}(X) \leq \frac{c^{2}}{4}
$$
HINT: One approach is to first argue that
$$
E\left[X^{2}\right] \leq c E[X]
$$
Then use this to show that
$$
\operatorname{Var}(X) \leq c^{2}[\alpha(1-\alpha)] \quad \text { where } \quad \alpha=\frac{E[X]}{c}
$$

Key Concepts

Optimization of Quadratic Functions

When an expression for variance is transformed into the form of a quadratic function, optimizing it helps in finding the maximum possible value under given constraints. In this context, a function like ?(1 - ?), where ? is a normalized expected value, reaches its maximum at ? = 1/2, leading to the conclusion that the variance is bounded by c^2/4. Recognizing and optimizing such quadratic forms is a common technique in deriving statistical inequalities.

Bounding Techniques for Random Variables

Bounding techniques involve using the known limits of a random variable to constrain other statistics, like its moments. For a variable that lies between two bounds, properties such as X^2 being less than or equal to a constant times X can be used to derive inequalities like E[X^2] ? cE[X]. This approach is crucial when establishing upper bounds on variance or other measures of dispersion.

Variance

Variance is a measure of the spread of a random variable around its mean. It is defined as the expectation of the squared difference between the variable and its mean, i.e., Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2. Understanding how variance is computed and its relationship with the second moment is key to establishing bounds on dispersion.

Expected Value

The expected value, or mean, of a random variable X is the long-run average of its values. It is central to many statistical concepts, serving as a point of reference in analyses such as the calculation of variance. In bounding problems, the expected value often provides a constraint that helps limit the possible values of the second moment.

Transcript

Please give Ace some feedback