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The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, also called the Gaussian curve. This distribution is symmetrical about its mean, showing that values near the mean are more frequent in occurrence than values far from the mean.
The properties of the normal distribution that make it universally applicable include:

• Mean, Median, Mode Equality: In a normal distribution, the mean, median, and mode are all equal, lying at the center of the distribution.
• Symmetry: The distribution is perfectly symmetrical about the mean, such that the area under the curve is distributed equally between the left and right sides.
• 68-95-99.7 Rule: Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
• Inflection Points: These points are located one standard deviation away from the mean on either side, marking where the curve changes concavity.

The behavior of a normally distributed variable makes it a fundamental building block in probability and statistics.

Linear transformation in probability involves changing a random variable using an equation of the form \(Y = aX + b\), where \(X\) is a random variable, \(a\) and \(b\) are constants. This transformation is crucial in changing the scale or location (mean) of a distribution.
Transformations impact the random variable as follows:

• Mean Change: The transformation alters the mean by a factor of \(a\) and translates it by \(b\). Thus, the new mean becomes \(a\mu + b\).
• Variance Change: Variance changes based on \(a^2\) and does not involve \(b\), resulting in a new variance of \(a^2\sigma^2\).

When a linear transformation is applied to a normally distributed variable, the result is another normal distribution. This is due to the transformation maintaining linearity, which implies the underlying distribution shape remains unchanged. The impact is merely on its parameters—mean and variance—transforming to reflect the new scale and shift.

Expected value, often denoted as \(E(X)\), provides the average or mean outcome of a random variable over numerous trials. It denotes the central tendency or 'long-term typical' value of the variable. For a normal distribution \(X\), the expected value is given by its mean \(\mu\).
Variance measures the spread of the random variable around the mean. It is represented by \(V(X)\) or \(\sigma^2\) for normal distributions. Variance quantifies variability, indicating how much the values deviate from the mean. A smaller variance means data is closely clustered around the mean, while a larger variance suggests more spread out data.For a linear transformation \(Y = aX + b\), the expected value and variance are computed as:

• Expected value of \(Y\): \(E(Y) = a\mu + b\)
• Variance of \(Y\): \(V(Y) = a^2\sigma^2\)

Both expected value and variance are crucial for understanding the behavior of transformed distributions, marking differences and consistencies in different contexts.

Temperature conversion often requires transforming normal distributions, especially for processes using different measurement systems like Celsius and Fahrenheit. The formula for converting Celsius to Fahrenheit is given by \(F = \frac{9}{5}C + 32\). This linear transformation enables understanding temperature variations across scales.
When the temperature in Celsius has a normal distribution, one can determine its distribution in Fahrenheit by applying the conversion formula. Suppose a temperature \(C\) in Celsius is normally distributed with mean \(\mu_C = 115\) and standard deviation \(\sigma_C = 2\). Converting to Fahrenheit, we use:

• Mean in Fahrenheit: \(E(F) = \frac{9}{5} \times 115 + 32 = 239\)
• Standard Deviation in Fahrenheit: \(\sigma_F = \frac{9}{5} \times 2 = 3.6\)

Thus, the temperature in Fahrenheit is normally distributed with a mean of 239 and a standard deviation of 3.6, reflecting transformed parameters due to the linear nature of the equation.