Problem 10 - Calculus for Biology and Medicine

Question

Suppose the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{cc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-1 & 0.2 \\ -0.5 & 0.25 \\ 0.1 & 0.1 \\ 0.5 & 0.1 \\ 1 & 0.35 \\ \hline \end{array}$$ Find and graph the corresponding distribution function \(F(x)\).

Step-by-Step Solution

Answer

The CDF \(F(x)\) is calculated as: \(F(-1)=0.2\), \(F(-0.5)=0.45\), \(F(0.1)=0.55\), \(F(0.5)=0.65\), \(F(1)=1.0\). The graph is a step function.

Step 1: Understand the Probability Mass Function

The given table represents probability mass function (PMF) values, where each value of \(x\) has an associated probability \( P(X = x) \). The PMF tells us the probability of each potential outcome for the discrete random variable \(X\).

Step 2: Calculate the Cumulative Distribution Function Values

The cumulative distribution function (CDF) \(F(x)\) is calculated by summing the probabilities for all values less than or equal to \(x\). We calculate \(F(x)\) at each distinct point in the PMF:- \(F(-1) = P(X = -1) = 0.2\)- \(F(-0.5) = P(X = -1) + P(X = -0.5) = 0.2 + 0.25 = 0.45\)- \(F(0.1) = P(X = -1) + P(X = -0.5) + P(X = 0.1) = 0.45 + 0.1 = 0.55\)- \(F(0.5) = P(X = -1) + P(X = -0.5) + P(X = 0.1) + P(X = 0.5) = 0.55 + 0.1 = 0.65\)- \(F(1) = P(X = -1) + P(X = -0.5) + P(X = 0.1) + P(X = 0.5) + P(X = 1) = 0.65 + 0.35 = 1.0\)

Step 3: Interpret and Graph the Cumulative Distribution Function

The CDF \(F(x)\) shows the probability that the random variable \(X\) is less than or equal to \(x\). To graph the CDF, plot each point \((x, F(x))\) calculated in Step 2, and connect these with horizontal lines, creating a step function:- Step from \(x = -1\) at \(F(x) = 0.2\)- Step from \(x = -0.5\) at \(F(x) = 0.45\)- Step from \(x = 0.1\) at \(F(x) = 0.55\)- Step from \(x = 0.5\) at \(F(x) = 0.65\)- Step from \(x = 1\) at \(F(x) = 1.0\)This graph will rise sequentially at each \(x\) value and remain constant between values.

Key Concepts

Probability Mass Function

The probability mass function (PMF) is an essential concept in probability theory. It is used to describe the probability distribution of a discrete random variable. When dealing with a discrete random variable, it only assumes specific, distinct values rather than any value in a continuous range. Each of these values is associated with a probability, summed up over all possible values to make 1. Let's break down the PMF in simpler terms:

• The PMF is denoted as \( P(X = x) \), showing the probability that a variable \( X \) takes a certain value \( x \).
• For any discrete values like \(-1\), \(-0.5\), \(0.1\), etc., the PMF provides the probability outcomes such as \(0.2, 0.25, 0.1\), and so on.
• The PMF is always a non-negative function, implying that \( P(X = x) \geq 0 \) for any \( x \).
• The sum of all probabilities from the PMF equals 1, ensuring a complete distribution of chances across all outcomes.

Understanding the PMF helps us grasp the underlying probabilities attached to each potential outcome of a discrete random variable, forming the groundwork for further exploration into cumulative distribution functions.

Discrete Random Variable

A discrete random variable is a type of random variable that takes on specific, distinct values. Unlike continuous variables that can occupy any value in an interval, discrete random variables are countable, often resulting from counting processes. Let's illuminate this concept further:

• Imagine rolling a six-sided die. The outcomes — 1 through 6 — are discrete values. Each side has a certain probability associated with it, forming the set of possible values.
• Discrete variables can include integers or specific categories, whereas continuous variables require measuring and can take any value within a range.
• The distribution of a discrete random variable is characterized by a PMF, which specifies the probabilities of these outcomes.
• A discrete random variable's probabilities, when visualized on a number line, often result in a "step" formation as each distinct outcome has a separate probability.

Understanding discrete random variables helps when dealing with situations where outcomes are distinct, defined, and separated, which is typical in many probability and statistical contexts. It’s crucial for accurately assessing and calculating probabilities of specific events.

Step Function

The concept of a step function is central when discussing cumulative distribution functions (CDFs) for discrete random variables. The CDF \( F(x) \) summarizes the probability that a random variable \( X \) is less than or equal to a specific value \( x \). Here's a closer look at how step functions work:

• A cumulative distribution function for a discrete random variable is a step function because it increases at certain points and remains constant between those points.
• Given the previous PMF examples, the CDF starts at 0 and "steps" up at each discrete value of \( x \) based on its cumulative probability. This means it will "jump" by specific amounts, reflecting the sum of probabilities up to that point.
• For example, if you have discrete points \( x = -1, -0.5, 0.1, 0.5, \) and \( 1 \), the CDF will rise dramatically at these values only, resulting in a graph that looks like steps or stairs.
• The CDF for a discrete variable is non-decreasing, meaning that as \( x \) moves forward, the value of \( F(x) \) either remains constant or increases but never decreases.

Understanding step functions is key in interpreting CDFs for discrete data, helping visualize how probabilities accumulate over the range of a variable, which is essential in many practical applications such as risk assessment and decision-making processes.