Problem 9 - Calculus for Biology and Medicine

Question

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{cc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-3 & 0.2 \\ -1 & 0.3 \\ 1.5 & 0.4 \\ 2 & 0.1 \\ \hline \end{array}$$ Find and graph the corresponding distribution function \(F(x)\).

Step-by-Step Solution

Answer

The CDF \(F(x)\) has jumps at \(-3, -1, 1.5,\) and \(2\), ending at 1.

Step 1: Understand the Problem

The problem provides a discrete random variable \(X\) with a probability mass function (pmf). Our goal is to find and graph the cumulative distribution function (CDF) \(F(x)\) for this random variable.

Step 2: Define Cumulative Distribution Function

Recall that the cumulative distribution function (CDF), \(F(x)\), is defined as the probability that the random variable \(X\) takes a value less than or equal to \(x\): \[ F(x) = P(X \leq x) \] This means we need to accumulate the probabilities from the probability mass function up to and including each point \(x\).

Step 3: Calculate CDF Values

Compute \(F(x)\) at each point \(x\) in the probability mass function:- For \(x < -3\), \(F(x) = 0\) because no probability is accumulated yet.- For \(x = -3\), \(F(-3) = P(X=-3) = 0.2\).- For \(-3 < x < -1\), \(F(x) = 0.2\).- For \(x = -1\), \(F(-1) = 0.2 + P(X=-1) = 0.2 + 0.3 = 0.5\).- For \(-1 < x < 1.5\), \(F(x) = 0.5\).- For \(x = 1.5\), \(F(1.5) = 0.5 + P(X=1.5) = 0.5 + 0.4 = 0.9\).- For \(1.5 < x < 2\), \(F(x) = 0.9\).- For \(x = 2\), \(F(2) = 0.9 + P(X=2) = 0.9 + 0.1 = 1\).- For \(x > 2\), \(F(x) = 1\).

Step 4: Graph the CDF

Plot the calculated CDF values:1. Draw the x-axis and mark the points -3, -1, 1.5, and 2.2. For \(x < -3\), the CDF is 0. This is a flat line at \(y=0\).3. At \(x = -3\), draw a jump to \(y=0.2\), and then a flat line until \(x = -1\).4. At \(x = -1\), draw a jump to \(y=0.5\), then a flat line until \(x = 1.5\).5. At \(x = 1.5\), draw a jump to \(y=0.9\), then a flat line until \(x = 2\).6. At \(x = 2\), draw a jump to \(y=1\), then continue flat since the CDF stabilizes at 1.

Key Concepts

Probability Mass Function

The probability mass function (PMF) is a fundamental concept when dealing with discrete random variables. It provides a way to represent the distribution of probabilities associated with all potential outcomes of a discrete random variable. With a PMF, you can easily determine the probability of each individual outcome by associating each possible value of the random variable with a specific probability. This is often presented in a table form. For example:

• If you have a dice, the PMF will assign a probability of 1/6 to each face, as this is a fair dice.
• In our problem, the PMF specifically assigns probabilities of 0.2, 0.3, 0.4, and 0.1 respectively to the values -3, -1, 1.5, and 2. These probabilities must sum up to 1 because they represent the entire distribution of possible outcomes for the variable.

Understanding how a PMF works is crucial to finding the cumulative distribution function which is built upon these probabilities.

Discrete Random Variable

A discrete random variable is a type of variable whose set of possible outcomes is countable and distinct. Unlike continuous random variables, discrete random variables can only take on specific values, often integers or fractions.Consider the following characteristics:

• Discrete random variables often represent counts or enumerated scenarios like the number of students in a class, the results of a die roll, or the outcomes under consideration in our assignment.
• They have a probability mass function (PMF) associated with them that provides the probability of occurrence for each possible value.

In the context of this exercise, our discrete random variable, denoted as \(X\), can take the values -3, -1, 1.5, and 2. These are distinct outcomes to which probabilities are assigned in the PMF. Understanding these values allows us to analyze cumulative probabilities accurately, leading to the cumulative distribution function (CDF).

Step Function Graph

A step function graph is a type of visualization essential for understanding cumulative distribution functions (CDFs) of discrete random variables. It vividly represents how probability accumulates over the range of possible values. Here are the key elements of a step function graph for a CDF:

• At each possible value of the discrete random variable, the graph makes a vertical "jump," which corresponds to the increase in cumulative probability at that point.
• Between these values, the graph remains flat, indicating that there is no increase in cumulative probability unless a defined outcome is reached.
• In our example, if you plot the CDF based on the data from the PMF, you'll see steps at -3, -1, 1.5, and 2, with flatness elsewhere.

These jumps and flats help interpret how a discrete random variable's outcomes stack to reach a total probability of 1. This graph not only conveys the CDF values but also makes interpreting probability shifts across values intuitive and clear.