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: $$ \begin{array}{rc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-3 & 0.2 \\ -1 & 0.3 \\ 1.5 & 0.4 \\ 2 & 0.1 \\ \hline \end{array} $$ Find and graph the corresponding distribution function \(F(x)\).

Step-by-Step Solution

Answer

The CDF is a step function with values: 0 for \( x < -3 \), 0.2 for \( -3 \leq x < -1 \), 0.5 for \( -1 \leq x < 1.5 \), 0.9 for \( 1.5 \leq x < 2 \), and 1 for \( x \geq 2 \). The graph is step-wise non-decreasing.

Step 1: Understand the Probability Mass Function (PMF)

The given table provides the PMF of the random variable \( X \). For each value of \( x \), the corresponding probability \( P(X = x) \) is listed. These probabilities add up to 1, verifying it's a valid PMF.

Step 2: Define the Cumulative Distribution Function (CDF)

The cumulative distribution function \( F(x) \) is defined as the probability that \( X \) is less than or equal to \( x \). Mathematically, \( F(x) = P(X \leq x) \).

Step 3: Calculate the CDF Values

Compute \( F(x) \) for each interval determined by the PMF:1. For \( x < -3 \), \( F(x) = 0 \).2. For \( -3 \leq x < -1 \), \( F(x) = P(X = -3) = 0.2 \).3. For \( -1 \leq x < 1.5 \), \( F(x) = P(X = -3) + P(X = -1) = 0.2 + 0.3 = 0.5 \).4. For \( 1.5 \leq x < 2 \), \( F(x) = P(X = -3) + P(X = -1) + P(X = 1.5) = 0.5 + 0.4 = 0.9 \).5. For \( x \geq 2 \), \( F(x) = 1 \).

Step 4: Construct the CDF Graph

Plot the CDF values step-wise without interpolating. For each interval specified in the previous step, the graph is horizontal since \( F(x) \) is constant over any interval between two of the PMF values. Use closed circles at the end of an interval and open circles at the beginning if they are not included in \( F(x) \).

Step 5: Verify Cumulative Probabilities

Ensure that the probability at the highest value of \( x \) equals 1, confirming all probabilities cumulatively sum up to 1. Also, check that the CDF is non-decreasing over the range of \( x \) values.

Key Concepts

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is a crucial concept in probability, especially when dealing with discrete random variables. It tells us the probability that a random variable, say \( X \), is less than or equal to a particular value \( x \). In simpler terms, it accumulates all the probabilities up to a certain point in the probability distribution table. This function is usually denoted as \( F(x) = P(X \leq x) \).

To compute the CDF from a probability mass function (PMF), you sum the probabilities from the smallest value up to the current value. For instance, if you have PMF values at certain discrete points, the CDF would remain flat between these points and will only jump at the values present in the PMF. For each interval between the scheduled points in the PMF, the CDF value is constant, reflecting the summed probabilities up to the beginning of that interval.

• If no less than or equal values precede \( x \), \( F(x) \) is zero.
• The value of \( F(x) \) will equal 1 once all values in the PMF are summed up.

CDF graphs are "step" functions due to this unchanged probability between specific data points. The graph is a series of horizontal lines above each interval determined by PMF values. These horizontal lines terminate with closed circles if the ending value is included or open circles if it is not.

Discrete Random Variable

A discrete random variable is a type of random variable that takes on distinct and separate values. Unlike continuous random variables, which can take on any value in a range, discrete random variables have fixed, countable values. For example, the roll of a dice results in a discrete set of numbers — 1 through 6.

In relation to probability distributions, a discrete random variable's probability mass function (PMF) provides the probability of each possible outcome specifically. Each outcome has a non-zero probability, and the sum of these probabilities is always 1. This is vital because it confirms that every potential outcome accounts for all possibilities of the random variable.

Understanding discrete random variables is essential, as they often form the basis in calculating further statistical measures, such as the mean and variance, which quantify the central tendency and dispersion of the variable's values, respectively. Knowing these measures gives more insight into the nature and behavior of the data described by the variable.

Probability Distribution Table

A Probability Distribution Table is an organized tabular representation showcasing how likely different outcomes are, for a specific random variable. The table typically associates each possible value of a discrete random variable with its corresponding probability. This tabular form helps to clearly understand the distribution and allows for a straightforward computation of functions like the CDF.

In the provided exercise, for instance, the probability distribution table is:

• For \( x = -3 \), \( P(X = -3) = 0.2 \).
• For \( x = -1 \), \( P(X = -1) = 0.3 \).


• For \( x = 1.5 \), \( P(X = 1.5) = 0.4 \).


• For \( x = 2 \), \( P(X = 2) = 0.1 \).

This table assigns each outcome of the random variable a specific probability, and the sum of these probabilities must equal 1. The probability distribution table forms the foundational data from which one constructs the PMF and subsequently, the CDF. Grasping this structure is important for any further analytical calculations involving discrete random variables.