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

a) \(\operatorname{var}(a X+b) = a^{2} \sum_{x}[x-E(X)]^{2} p(x)\); b) \(\operatorname{var}(a X+b) = a^2 \operatorname{var}(X)\).

Step 1: Understanding Variance

The variance of a random variable \( X \) is defined as \( \operatorname{var}(X) = \mathbb{E}[(X - \mathbb{E}(X))^2] \), where \( \mathbb{E}(X) \) is the expected value of \( X \). For a transformed random variable \( aX + b \), we need to find \( \operatorname{var}(aX + b) \).

Step 2: Applying the Variance Definition

Start by expressing the variance of \( aX + b \):\[ \operatorname{var}(aX + b) = \mathbb{E}[(aX + b - \mathbb{E}(aX + b))^2] .\]

Step 3: Simplify the Expression

Since \( \mathbb{E}(aX + b) = a\mathbb{E}(X) + b \), substitute into the variance expression:\[ \operatorname{var}(aX + b) = \mathbb{E}[(aX + b - (a\mathbb{E}(X) + b))^2] \]Simplify to:\[ \operatorname{var}(aX + b) = \mathbb{E}[(aX - a\mathbb{E}(X))^2] \].

Step 4: Factor Out Constants

With the expression \((aX - a\mathbb{E}(X))^2 = a^2(X - \mathbb{E}(X))^2\), factor out \( a^2 \):\[ \operatorname{var}(aX + b) = a^2\mathbb{E}[(X - \mathbb{E}(X))^2] \].

Step 5: Express in Terms of Probability Function

Since \( \mathbb{E}[(X - \mathbb{E}(X))^2] = \sum_x (x - \mathbb{E}(X))^2 p(x) \), substitute into the expression:\[ \operatorname{var}(aX + b) = a^2 \sum_{x} (x - \mathbb{E}(X))^2 p(x) \]This satisfies part (a).

Step 6: Concluding Part (b)

Recall that \( \operatorname{var}(X) = \sum_{x} (x - \mathbb{E}(X))^2 p(x) \). Thus:\[ \operatorname{var}(aX + b) = a^2 \operatorname{var}(X) \]This satisfies part (b).

Key Concepts

Discrete Random Variable

A discrete random variable is a type of random variable that can take on a finite or countably infinite set of values, such as an integer or a whole number. These are used to model outcomes that are distinct and separate. For example, the number of heads in a series of coin tosses or the number of students in a classroom could be modeled by discrete random variables.

The probabilities associated with each possible outcome of a discrete random variable are summed up to one, as all possible events are accounted for within its finite range. This means that the sum of probabilities taken over all possible outcomes equals one.

• Discrete random variables have specific values.
• They model countable outcomes.
• They sum up to a probability of one.

This distinct nature of values makes them different from continuous random variables, which can assume an uncountably infinite number of values, such as any real number between intervals.

Expected Value

The expected value of a discrete random variable is essentially a measure of the 'central' value that the variable tends toward over a large number of observations. It provides a weighted average of all possible values that the random variable can take, where the weights are the probabilities of each outcome.

Mathematically, it is calculated as \(\mathbb{E}(X) = \sum_{x} x p(x) \), where \(x\) denotes a possible value of the random variable \(X\) and \(p(x)\) the probability of \(X\) taking the value \(x\). Expected value is an important concept in probability and helps in understanding the average outcome in the long term.

• It represents the average outcome.
• It weights each possible value by its probability.
• It's a central measure of a random variable.

Expected value plays a role in various domains such as economics, finance, and decision theory, guiding predictions and strategic decisions.

Transformation of Variables

Transformation of variables refers to modifying a random variable to create a new one. Often, this involves applying mathematical changes, such as scaling or shifting the variable. For example, if you have \(X\), and you transform it to \(aX + b\), you have scaled \(X\) by \(a\) and shifted it by \(b\).

This type of transformation is beneficial in scenarios where you want to adjust the characteristics of a distribution, such as its variance or mean. When dealing with transformations, it's crucial to understand how these affect properties like variance. In particular, transforming \(X\) linearly as \(aX + b\) scales the variance by \(a^2\), while shifts like adding \(b\) don't affect variance.

• Linear transformations include scaling and shifting.
• Scaling affects variance; shifting does not.
• Useful for adjusting mathematical properties.

Transformation is a common technique in statistical analysis, especially in data normalization and standardization.

Probability Functions

Probability functions define the likelihood of different outcomes of a random variable. For discrete variables, this is termed the probability mass function (PMF). It assigns probabilities to each possible outcome so that the sum of these probabilities equals one.

The PMF for a random variable \(X\) is denoted as \(p(x) = P(X = x)\). It provides a complete description of the distribution of \(X\). Understanding the probability function is crucial, as it allows us to compute other essential properties, like the expected value and variance.

• Defines outcome probabilities for random variables.
• Sum of all probabilities is 1.
• Essential for computing other statistics like expected value and variance.

These functions play a crucial role in statistical methods, enabling prediction and understanding of random phenomena. Moreover, they help in visualizing data distributions and comparing different datasets.