Problem 25 - 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 $$E(a X+b)=\sum_{x}(a x+b) p(x)$$ (b) Use your result in (a) and the rules for finite sums to conclude that $$E(a X+b)=a E(X)+b$$ 26\. 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

(a) Showed \(E(aX+b) = aE(X) + b\). (b) Derived \(\operatorname{var}(aX+b) = a^2 \operatorname{var}(X)\).

Step 1: Understand Expectation Formula

The expectation of a discrete random variable is given by \(E(X) = \sum_{x} x p(x)\). For a transformed variable \(aX + b\), the expectation becomes \(E(aX + b) = \sum_{x} (a x + b) p(x)\). We need to show this formula in the next step.

Step 2: Derive Expectation for \(aX + b\)

Substitute \(a x + b\) into the expectation formula: \[E(a X + b) = \sum_{x} (a x + b) p(x) = \sum_{x} a x p(x) + \sum_{x} b p(x).\]Since \(b\) is a constant, it can be factored out:\[E(a X + b) = a \sum_{x} x p(x) + b \sum_{x} p(x).\]

Step 3: Simplify the Expectation Equation

Recognize that \(\sum_{x} x p(x) = E(X)\), the expectation of \(X\), and \(\sum_{x} p(x) = 1\) by definition of probability mass function. Thus:\[E(a X + b) = a E(X) + b.\]

Step 4: Define Variance Transformation Formula

By definition, variance is \( \operatorname{var}(X) = E([X - E(X)]^2) = \sum_{x} [x - E(X)]^2 p(x) \). We need to compute \( \operatorname{var}(aX + b) \).

Step 5: Compute Variance of \(aX + b\)

Form the expression:\[ \operatorname{var}(aX + b) = E([(aX + b) - E(aX + b)]^2). \]Substituting the expectation \(E(aX + b) = a E(X) + b\), we get:\[= E([aX + b - (aE(X) + b)]^2) = E([a(X - E(X))]^2).\]Since \(a^2\) is a constant factor:\[= a^2 E([X - E(X)]^2).\]

Step 6: Simplify Variance Expression

Recognize that \(E([X - E(X)]^2) = \operatorname{var}(X)\), so the variance simplifies to:\[ \operatorname{var}(aX + b) = a^2 \operatorname{var}(X). \]Verify using the properties that constant factors can be pulled out of the variance function.

Key Concepts

Expectation of a Random Variable

The expectation, or expected value, of a discrete random variable is a fundamental concept that essentially captures the average value of the variable, weighted by its probabilities. When dealing with a discrete random variable \(X\), the expectation is calculated with the formula \(E(X) = \sum_{x} x \cdot p(x)\), where \(p(x)\) is the probability mass function.

• The expectation sums up all possible values of the variable, each multiplied by its probability.
• In mathematical terms, this is a weighted sum that gives us the mean or average of the potential outcomes.

When variables are transformed, such as \(aX + b\) where \(a\) and \(b\) are constants, the calculation for expectation becomes \(E(aX + b) = aE(X) + b\). Here’s why this makes sense:

• First, you substitute \(aX + b\) into the expectation, leading to \(\sum_{x} (ax + b) p(x)\).
• Breaking it down further: \(\sum_{x} ax \cdot p(x) + \sum_{x} b \cdot p(x)\).

Because \(b\) is constant and doesn't change with \(x\), it factors out, and since \(\sum_{x} p(x) = 1\), the final formula simplifies to \(aE(X) + b\). This formula is crucial for understanding linear transformations of random variables and their expected outcomes.

Variance of a Random Variable

Variance measures how much the values of a random variable differ from the expected value, helping us understand the variable's spread or variability. Calculating the variance \( \operatorname{var}(X) \) of a discrete random variable involves the formula: \(E([X - E(X)]^2) = \sum_{x} [x - E(X)]^2 \cdot p(x)\).

• This quantifies the average squared deviation of each possible value from the mean, weighted by its probability.
• Variance is especially used to understand the dispersion and can be thought of as how volatile or consistent the variable is.

When applying transformations like \(aX + b\), the variance transformation follows these steps:

• Consider the expression for variance, \(\operatorname{var}(aX + b) = E([(aX + b) - E(aX + b)]^2)\).
• Using \(E(aX + b) = aE(X) + b\), substitute into the variance to get \(E([a(X - E(X))]^2)\).

Notice that \(a^2\) can be pulled out as a constant, leaving \(= a^2 E([X - E(X)]^2)\), which simplifies further to \(a^2 \cdot \operatorname{var}(X)\). This shows how linear scaling affects variability, magnifying it by a factor of \(a^2\). It elegantly demonstrates how transformations impact the spread of probability distributions.

Probability Mass Function

The probability mass function (PMF) is a key element when handling discrete random variables. It assigns probabilities to each possible outcome of a random variable, symbolized as \(p(x) = P(X = x)\).

• Each probability \(p(x)\) must satisfy \(0 \leq p(x) \leq 1\).
• The total sum of all probabilities is always 1, which maintains the logical foundation of probability.

The PMF provides a complete description of the random variable's distribution by listing each value and its corresponding probability. When using PMFs in calculations like expectation or variance, they ensure that each outcome’s contribution is appropriately weighed. For instance:

• In expectation calculations, \(E(X) = \sum_{x} x \cdot p(x)\), the PMF ensures all outcome-weightings sum to a meaningful average.
• For variance, \(\operatorname{var}(X) = \sum_{x} [x - E(X)]^2 \cdot p(x)\), the PMF again helps weigh deviations by their likelihood.

Understanding the PMF is fundamental for interpreting any discrete random variable since it forms the basis of further statistical computations and analyses.