Problem 26 - Calculus for Biology and Medicine

Question

Assume that \(X\) is a discrete random variable with finite range, and set $$ p(x)=P(X=x) $$ (a) Show that $$ \operatorname{var}(a X+b)=a^{2} \sum_{x}[x-E(X)]^{2} p(x) $$ (b) Use your result in (a) and the rules for finite sums to conclude that $$ \operatorname{var}(a X+b)=a^{2} \operatorname{var}(X) $$

Step-by-Step Solution

Answer

\( \operatorname{var}(aX+b) = a^2 \operatorname{var}(X) \). Variance scales by \( a^2 \).

Step 1: Understand the Variance Formula

The variance of a random variable \( X \) is defined as \( \operatorname{var}(X) = E[(X - E(X))^2] \). This represents the expectation of the squared deviations from the mean \( E(X) \). When dealing with a transformed variable \( aX+b \), the variance becomes \( \operatorname{var}(aX+b) = E[((aX+b) - E(aX+b))^2] \).

Step 2: Simplify the Expectation of the Transformed Random Variable

Calculate \( E(aX+b) = aE(X) + b \). Now, substitute this into the variance formula: \( \operatorname{var}(aX+b) = E[((aX+b) - (aE(X) + b))^2] \). Simplifying inside the expectation gives: \( \operatorname{var}(aX+b) = E((a(X-E(X)))^2) \).

Step 3: Simplify Further Using algebra

Factor out \( a^2 \) from the squared term: \( \operatorname{var}(aX+b) = E(a^2(X-E(X))^2) = a^2 E((X-E(X))^2) \). Since \( E((X-E(X))^2) = \operatorname{var}(X) \), we have \( \operatorname{var}(aX+b) = a^2 \operatorname{var}(X) \).

Step 4: Conclusion

(a) We derived that \( \operatorname{var}(aX+b) = a^2 \sum_x [x-E(X)]^2 p(x) \). (b) Using the property of expectations, we confirmed that \( \operatorname{var}(aX+b) = a^2 \operatorname{var}(X) \), showing that variance scales with the square of the coefficient \( a \) applied to \( X \).

Key Concepts

Discrete Random Variable

In probability and statistics, a discrete random variable is a type of variable that takes on distinct, separate values. These are countable values, often resulting from a specific sample space. For example, rolling a die results in a discrete random variable because the outcome is limited to 1 through 6.

Discrete random variables are characterized by a probability mass function, denoted by \(p(x) = P(X=x)\), which gives the probability that the random variable \(X\) is exactly equal to \(x\). One key aspect to remember is that the sum of probabilities for all possible values should equal 1.

• Each potential outcome has a probability assigned to it.
• The range of the variable is finite or countably infinite.

Expectation

Expectation, or the expected value, of a discrete random variable is akin to a mean or average value one might expect for an occurrence. It is a fundamental concept when considering random variables and is computed as the sum of all possible values of the variable, each multiplied by its probability.

Mathematically, for a discrete random variable \(X\) with range \(x_1, x_2, ..., x_n\), the expectation \(E(X)\) is computed as:\[E(X) = \sum_{x} x \cdot p(x)\]
This expectation represents the center or 'balancing point' of the distribution of the variable. Understanding this allows for further computations related to transformations and variance, setting the base for more complex analyses.

Algebraic Simplification

Algebraic simplification is a key technique in dealing with operations involving random variables, especially while manipulating equations to solve for expectations or variances. When we transform a random variable, like turning \(X\) into \(aX+b\), there is a need to redefine expressions algebraically for clarity and solution.

For instance, transforming the expectation of \(aX + b\) can be computed using:\[E(aX + b) = aE(X) + b\]This arises from algebraic principles that allow us to independently manage linear transformations.

• Simplifying these transformations is crucial for subsequent calculations involving variance.
• By breaking down transformations into smaller steps, complex problems become more manageable.

Variance Scaling

Variance scaling is an important concept when studying how transformations affect the variability of random variables. When a random variable is scaled and shifted, the variance behaves predictably, particularly with scaling.
The key principle is that if a random variable \(X\) is scaled by a factor \(a\) (i.e., transformed into \(aX+b\)), the variance of this new variable \(\text{var}(aX+b)\) can be calculated as:\[\text{var}(aX+b) = a^2 \text{var}(X) \]
This shows that the variance scales with the square of the factor \(a\). This is an essential insight because it reveals that while shifting \(b\) does not affect variability, scaling \(X\) by \(a\) significantly changes the spread or dispersion of values around the mean.

• Transformations like this maintain the linearity of expectation while modifying variance drastically with the factor squared.
• Understanding this helps predict outcomes under variables transformations and influences decisions in data modeling.