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A continuous random variable can take on infinitely many values within a given range. Unlike a discrete random variable, which can only assume specific values, a continuous random variable could be any value along a continuum. For instance, think of measuring the exact height of people in a room. You could get values like 5.833 feet, 6.421 feet, and so on. Since these values cover a spectrum, you need a different approach to work with probabilities.

Probabilities for continuous random variables are defined over intervals, not individual points. This is because, in theory, the probability of a continuous random variable taking any exact value is zero. Instead, you find probabilities over specified ranges. For instance, the probability that a height falls between 5.5 and 6.5 feet.

In the context of a standard normal distribution, a continuous random variable often denoted by the letter 'Z', represents standard scores (Z-scores). The probability that this score falls within a specified range can be found using statistical tools like the Z-table.

The Z-table is a tool used to find the cumulative probabilities associated with standard normal distribution values, also known as Z-scores. Z-scores represent how many standard deviations an element is from the mean of the distribution. To use the Z-table effectively, follow these steps:

• Identify the Z-score value for which you need the cumulative probability. The Z-score is calculated as \(z = \frac{(X - \text{mean})}{\text{standard deviation}}\).
• Locate the Z-score in the Z-table. The table usually lists scores to two decimal places; one digit on the side and the second along the top.
• Read off the cumulative probability corresponding to that Z-score. This value represents the area under the normal curve to the left of the specified Z-score.

For example, from the problem, the Z-scores are -2.01 and 0. According to the Z-table, the cumulative probability for \(z = -2.01\) is approximately 0.0222, and for \(z = 0\), it is 0.5000.

These cumulative probabilities are used to determine the likelihood that the random variable falls within a certain range.

Cumulative probability represents the probability that a random variable takes on a value less than or equal to a specified value. When using the Z-table for a standard normal distribution, the cumulative probability is the area under the curve to the left of a particular Z-score. This gives the probability that Z will be less than or equal to that Z-score.

Let's break down the steps using the exercise example:

• First, identify the Z-scores: -2.01 and 0.
• Find the cumulative probabilities corresponding to these Z-scores using the Z-table: 0.0222 for \(z = -2.01\) and 0.5000 for \(z = 0\).
• The cumulative probability for any Z-score indicates the area under the normal distribution curve to the left of that Z-score.

To find the probability that a variable falls between \(z = -2.01\) and \(z = 0\), subtract the cumulative probability at \(z = -2.01\) from the cumulative probability at \(z = 0\). Thus, we get:

\(\text{Probability} = \text{Cumulative Probability at } z=0 - \text{Cumulative Probability at } z=-2.01\)

Which is

\(\text{Probability} = 0.5000 - 0.0222 = 0.4778\).

This calculation tells you the likelihood that the random variable falls between the Z-scores given.