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: $$ \begin{array}{rl} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-1 & 0.2 \\ -0.5 & 0.25 \\ 0.1 & 0.1 \\ 0.5 & 0.1 \\ 1 & 0.35 \\ \hline \end{array} $$ Find and graph the corresponding distribution function \(F(x)\).

Step-by-Step Solution

Answer

The CDF \(F(x)\) has jumps at each \(x\): \((-1,0.2), (-0.5,0.45), (0.1,0.55), (0.5,0.65), (1,1.0)\).

Step 1: Understanding Probability Mass Function (PMF)

The probability mass function (PMF) provides the probability that a discrete random variable is exactly equal to some value. From the provided table, we get the PMF of random variable \(X\), where each value of \(x\) has its corresponding probability.

Step 2: Definition of Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF), \(F(x)\), of a random variable \(X\), gives the probability that \(X\) will take a value less than or equal to \(x\). Mathematically, it is defined by \(F(x) = P(X \leq x)\).

Step 3: Calculating the CDF at Each Value

Using the PMF, calculate the CDF at each value of \(x\):- \(F(-1) = P(X \leq -1) = 0.2\)- \(F(-0.5) = P(X \leq -0.5) = 0.2 + 0.25 = 0.45\)- \(F(0.1) = P(X \leq 0.1) = 0.45 + 0.1 = 0.55\)- \(F(0.5) = P(X \leq 0.5) = 0.55 + 0.1 = 0.65\)- \(F(1) = P(X \leq 1) = 0.65 + 0.35 = 1.0\)

Step 4: Graph the CDF

Plot the CDF. Start at \(F(-1) = 0.2\), jump to \(F(-0.5) = 0.45\), then to \(F(0.1) = 0.55\), \(F(0.5) = 0.65\), and finally \(F(1) = 1.0\). The graph is a step function with jumps corresponding to the probability at each value of \(x\).

Key Concepts

Probability Mass Function

A Probability Mass Function (PMF) is an essential concept in probability theory, especially when dealing with discrete random variables. It gives the probability that a discrete random variable is equal to a specific value. So, for a discrete random variable like our example, the PMF essentially maps every possible value of the random variable to its probability of occurrence.

A discrete random variable can only take on certain values. This means the PMF will be a list or a table of these values and their corresponding probabilities. In our exercise, the PMF is provided in the table format, with each value of \(x\) and the probability \(P(X = x)\). This makes it very straightforward to see which values \(X\) can take, and how likely each one is.

• The probabilities listed must sum up to 1, which is a basic property of PMFs.
• Each probability must lie between 0 and 1, inclusive.

In this setup, the PMF helps us compute the Cumulative Distribution Function (CDF), since it provides the base data needed for such calculations.

Discrete Random Variables

Discrete random variables are those that take on a countable number of distinct values. Unlike continuous random variables that can take on an infinite number of values within a given range, discrete random variables have specific, isolated points where they can land.

In probability and statistics, discrete random variables are very useful because they can represent outcomes that are naturally separate. For example, the number of students in a class or the result of rolling a die are naturally discrete events.

• They are associated with discrete probability distributions like the PMF.
• Values are often represented with an accompanying probability that assigns the probability of that specific outcome occuring.

In the example exercise, \(X\) is a discrete random variable. Each value \(x\) it takes has been assigned a probability, showcasing the nature of the PMF for a discrete set.

Step Function

A step function is a mathematical function that changes abruptly from one constant value to another. When we talk about a cumulative distribution function (CDF) for a discrete random variable, the graph of this function is a classic example of a step function.

The CDF maps each possible value \(x\) of the random variable to the probability that the variable is less than or equal to \(x\). In our exercise, after computing \(F(x)\) for each \(x\) using the PMF, the resulting CDF was plotted as a step function.

• The step function is non-decreasing, meaning it either remains constant or increases as \(x\) increases.
• Each jump in the CDF plot corresponds to a value \(x\) with a positive probability mass in the PMF, indicating an increase in cumulative probability.

Graphing the CDF visually demonstrates these jumps, with each step in the function at a value \(x\) showing the cumulative probability accumulating based on the PMF.