Normal Distribution

Approximation

Models the number of successes \(X\) in \(n\) independent Bernoulli trials, each with success probability \(p\).

\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]

• Given,
  • \(X \sim \text{Binomial}(n, p)\)
  • Mean: \(\mu = np\)
  • Variance: \(\sigma^2 = np(1 - p) = npq\)

Normal Approximation

For large \(n\), the binomial distribution approximates a normal distribution:

\[ X \approx \mathcal{N}(np, npq) \]

• Bell curve shape
• Approximation becomes accurate for \(n \gtrsim 30\) and moderate \(p\)

• Bernoulli Trials:

  Single trial with two outcomes → success (probability \(p\)) or failure (probability \(1 - p\))

Equation

The probability density function (PDF) of a normal (Gaussian) distribution with mean \(\mu\) and standard deviation \(\sigma\) is:

\[ P(X = x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right) \]

Where:

• \(\mu = n \cdot p\) is the mean (expected value),
• \(\sigma^2 = n \cdot p \cdot q\) is the variance,
• \(\sigma = \sqrt{n \cdot p \cdot q}\) is the standard deviation,
• \(q = 1 - p\) is the probability of failure.

This formula gives a continuous approximation of the binomial distribution when \(n\) is large and \(p\) is not too close to 0 or 1.

Descriptive Statistics

Type	Formula
Support	\(x \in (-\infty, \infty)\)
Mean	\(\mathbb{E}[X] = \mu\)
Variance	\({Var}(X) = \sigma^2\)
Standard Deviation	\(\sigma\)
Probability Density Function (PDF)	\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\)
Cumulative Distribution Function (CDF)	\(F(x) = \mathbb{P}(X \leq x) = \int_{-\infty}^x f(t) dt\)

Intuition and Use

• Why Approximate Binomial with Normal?
  • Binomial becomes computationally intensive for large \(n\)
  • Normal is continuous and easy to compute probabilities for \((e.g., P(20 ≤ X ≤ 50))\). This would be a monster to compute using Binomial distribution.
  • Better for analysis and visualization
  • Normal distributions have well-known properties for sums, differences, and linear combinations. This makes complex probability problems much more manageable.
• When Does Approximation Work Well?
  • \(n\) is large (rule of thumb: \(n \geq 30\) or so). The distributions agree well with a lot of overlap.
  • \(p\) not too close to 0 or 1 (i.e., \(np(1 - p)\) not small)
  • Fails when \(p\) or \(q\) is too small, may predict impossible values (like negative successes)
• Central Limit Theorem (CLT):
  • Sum of many i.i.d. random variables (of any distribution with finite variance) converges to a normal distribution
  • Binomial is a sum of Bernoulli trials → classic CLT case

Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

• PDF describes the probability density at each point xx; area under the curve between two points gives the probability that \(X\) lies within that range.
• CDF is the integral of the PDF from \(\int_{-\infty}^x f(x) dx\), representing the probability that \(X \leq x\).
• The standard normal distribution (\(\mu=0, \sigma=1\)) is the canonical form, often denoted \(\Phi(x)\) for the CDF.

Calculating Probabilities

• Probability \(P(a \leq X \leq b) = \int_a^b f(x) dx = \Phi(b) - \Phi(a)\), where \(f(x)\) is the PDF.
• CDF values can be found using lookup tables historically, now easily computed with Python libraries like SciPy.

Applying Normal Approximation to Binomial

• Example: Flip a fair coin 400 times \((n=400, p=0.5)\):
  • Mean \(\mu = np = 200\)
  • Variance \(\sigma^2 = np(1-p) = 100\)
  • Standard deviation \(\sigma = 10\)
• Calculating let’s say getting # of heads between → \(P(190 \leq X \leq 230)\) binomially requires summing 51 probabilities, cumbersome to compute directly.
• Using normal approximation, calculate \(\Phi(230) - \Phi(190)\) to get probability efficiently.