Problem 19 - Calculus and its applications

Question

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-2.45 \leq x \leq-1.24) $$

Step-by-Step Solution

Answer

The probability is 0.1004.

Step 1: Understanding the Probability Range

We need to find the probability that a standard normal variable, \(x\), falls within the interval \([-2.45, -1.24]\). This involves using the standard normal distribution table (often called Table A) to find the cumulative probability for \(x\) at \(z = -2.45\) and \(z = -1.24\).

Step 2: Find Cumulative Probability for z = -2.45

Using the standard normal distribution table, locate \(z = -2.45\). The value given in the table is the cumulative probability up to \(z = -2.45\), which we'll denote as \(P(x \leq -2.45)\). From the table, \(P(x \leq -2.45) = 0.0071\).

Step 3: Find Cumulative Probability for z = -1.24

Similarly, use the table to find the cumulative probability for \(z = -1.24\). This probability is \(P(x \leq -1.24)\), and from the table, \(P(x \leq -1.24) = 0.1075\).

Step 4: Calculate the Probability of the Range

The probability of \(-2.45 \leq x \leq -1.24\) can be found by subtracting the cumulative probability at \(z = -2.45\) from the cumulative probability at \(z = -1.24\). Thus, \(P(-2.45 \leq x \leq -1.24) = P(x \leq -1.24) - P(x \leq -2.45) = 0.1075 - 0.0071 = 0.1004\).

Key Concepts

Cumulative Probability

Cumulative probability is a key concept when dealing with random variables and distributions. It is the probability that a random variable takes a value less than or equal to a certain point. For any random variable, the cumulative probability is represented as a function, known as the cumulative distribution function (CDF). If we denote our random variable as \( x \), then the CDF, \( P(x \leq a) \), gives us the probability that \( x \) is less than or equal to some specific value \( a \).

• It accumulates probability from the left of the distribution up to point \( a \).
• In a continuous distribution like the standard normal, the CDF is smooth and continuous.
• It helps determine intervals and specific probabilities within those intervals.

In our exercise, we are interested in the cumulative probabilities at two points, \( z = -2.45 \) and \( z = -1.24 \). Using these cumulative probabilities allows us to find the probability of the random variable \( x \) falling within the interval \(-2.45 \leq x \leq -1.24\). This is done by subtracting the smaller cumulative probability at \( z = -2.45 \) from the larger cumulative probability at \( z = -1.24 \).
This subtraction is a powerful technique that provides the probability for values falling within a range of interest.

Standard Normal Distribution Table

The standard normal distribution table, often referenced as Table A, is a vital tool for anyone working with normally distributed variables. It displays cumulative probabilities of a standard normal random variable according to the z-scores. Here are some essential points about it:

• The table includes values for \( z \), ranging typically from negative to positive numbers.
• Each entry in the table provides the cumulative probability that a normally distributed random variable is less than or equal to the corresponding \( z \) value.
• Considering symmetry, negative \( z \) values help determine probabilities in the left tail of the distribution.

The task at hand involved using this table to pinpoint the cumulative probabilities for \( z = -2.45 \) and \( z = -1.24 \). Using these table values (0.0071 and 0.1075, respectively), we were able to calculate the probability of what falls between these two values in the standard normal distribution.
Having such a table allows for straightforward calculations, as long as you understand how to read and interpret it effectively. It simplifies many statistical analyses by providing a quick reference for probability assessments at specific z-scores.

Random Variable

A random variable is an essential concept in probability and statistics. It represents a variable whose possible values are numerical outcomes of a random phenomenon.

• Random variables can be discrete or continuous. In this exercise, we focus on a continuous random variable which can take infinitely many values in a given range.
• Each outcome of a random variable is associated with a probability, contributing to overall probabilistic analyses.
• In terms of probability distribution, a random variable is described by all its possible values and their corresponding probabilities.

When discussing the standard normal distribution, the random variable \( x \) is normally distributed with a mean of 0 and a standard deviation of 1. This is crucial because it directly influences how we interpret probabilities and use the standard normal distribution table.
Understanding random variables lays the groundwork for comprehending more complex statistical themes. They provide a foundation for modeling real-world phenomena where outcomes are subject to randomness and uncertainty.