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A continuous random variable is a variable that can take an infinite number of possible values. Unlike discrete random variables, which can only take specific, distinct values, a continuous random variable is more fluid. Its values are not restricted to certain points on the number line. For example, if you think about measuring the weight of apples, the weight could be 150 grams or 150.346 grams, or really any value within the range. This flexibility in values makes continuous random variables suitable for measurements like time, height, or in this case, the variable \(X\) with a normal distribution.

Continuous random variables are characterized by a probability density function. This function describes the relative likelihood of the variable taking on a value within a particular range. Although, the probability at any single, precise point is actually zero because there are infinitely many possible points. What we look at is the probability over a range of values.

The mean of a continuous random variable is the average or expected value of the distribution. In the context of a normal distribution, this is also the central point of the distribution's symmetry. For a normally distributed variable \(X\), the mean \(\mu\) is what you would expect to be the most likely outcome. All the statistical data around it tends to balance out to this central number. In our exercise, the mean is given as \(50.5\).

The mean is crucial because in a symmetric distribution like the normal distribution, it tells you exactly where the peak of the distribution curve is. This point of symmetry is vital when calculating probabilities for ranges on either side. If you think of the distribution as a mountain, the mean is the highest peak right in the center.

Probability is a measure that helps us understand how likely an event is to happen. It ranges from 0 to 1, where 0 means an event will not happen, and 1 means it will definitely happen. When working with continuous random variables and their probability density functions, we're often interested in finding the probability of \(X\) falling within a specific interval.

In the given problem, it describes the probability that \(X\) takes a value less than 54, which is \(0.76\). This tells us that there is a 76% chance \(X\) is less than 54. Due to the entire normal distribution curve totaling an area of 1, you can easily find other probabilities, such as \(P(X \geq 47)\) by using symmetry and the complement rule.

• \(P(X<54) = 0.76\)
• \(P(X>47) + P(X \leq 47) = 1\)
• \(P(X>47) = 0.76\)

The normal distribution is known for its perfect symmetry around the mean. This symmetry allows for convenient probability calculations. If you know the mean \(\mu\), you're aware that the curve looks like a bell, perfectly mirrored from side to side across this mean.

Thanks to this symmetry, you can draw important conclusions about probabilities. For an event such as \(P(X < 54) = 0.76\), the natural symmetry implies that \(P(X > 47)\) can also be \(0.76\), as seen in our solution. The other half of the curve complements these calculations to ensure the total probability remains 1.

• The shape of the curve ensures that probabilities of reaching below or exceeding certain values balance out around the mean.
• Visualization of this symmetry can often aid in understanding; drawing the curve and shading respective probabilities on either side helps conceptualize this property.