To create symbolic expressions in line with the steps you provided, I'll show you how it can be done using symbolic variables and manipulation of expressions. I will explain each step carefully below, assuming you are using a platform like MATLAB or Python's SymPy library.

Step 1: Create Symbolic Variables

We first define the symbolic variables $x$, $m$, and $s$, which correspond to the mathematical symbols $x$, $\mu$ (mean), and $\sigma$ (standard deviation), respectively.

In Python's SymPy:

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from sympy import symbols

# Define symbolic variables x, m, s
x, m, s = symbols('x m s')

In MATLAB:

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syms x m s

Step 2: Create Symbolic Expression expr1

Next, we create the symbolic expression $\frac{x - m}{s}$, which is a common standardization formula used in statistics.

In Python's SymPy:

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# Define the expression expr1
expr1 = (x - m) / s

In MATLAB:

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expr1 = (x - m) / s;

Step 3: Create Symbolic Expression expr2

Now, using expr1, we create another expression $\left(\frac{x - m}{s}\right)^2$, which is the square of the standardized expression.

In Python's SymPy:

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# Define expr2 as the square of expr1
expr2 = expr1 ** 2

In MATLAB:

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expr2 = expr1 ^ 2;

Step 4: Create Symbolic Expression expr3

Finally, using expr2, we define the third expression $e^{-\frac{1}{2} \left( \frac{x - m}{s} \right)^2}$, which is commonly used in the Gaussian distribution function.

In Python's SymPy:

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from sympy import exp

# Define expr3 as the exponential of (-1/2 * expr2)
expr3 = exp(-1/2 * expr2)

In MATLAB:

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expr3 = exp(-1/2 * expr2);

Summary of Expressions

1. $\text{expr1} = \frac{x - m}{s}$
2. $\text{expr2} = \left( \frac{x - m}{s} \right)^2$
3. $\text{expr3} = e^{-\frac{1}{2} \left( \frac{x - m}{s} \right)^2}$

Would you like more details about any of these steps or further explanation?

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Here are 5 related questions that expand on this topic:

1. How can you simplify symbolic expressions involving multiple variables in Python or MATLAB?
2. How can symbolic expressions be differentiated with respect to $x$?
3. What is the significance of $\left(\frac{x - m}{s}\right)^2$ in statistics, and how is it used in probability distributions?
4. How do you numerically evaluate a symbolic expression for specific values of $x$, $m$, and $s$?
5. How can symbolic integration be applied to $expr3$ to find the integral of the Gaussian distribution?

Tip: Symbolic computation is very useful for deriving analytical solutions. Always try to simplify expressions when possible to avoid errors in further computations.