Notes on Uniform, Normal Distributions, Z-scores, and CDFs

Uniform distribution

• Concept: density is constant across its support; probability is proportional to area under the density over an interval.
• Example distribution: a uniform distribution on $[0,\,10]$
  • Probability density function (PDF):
    $f(x) = \frac{1}{10}, \quad 0 \le x \le 10$
  • Cumulative distribution function (CDF):
    F(x) = \begin{cases}0, & x < 0 \[6pt]\frac{x}{10}, & 0 \le x \le 10 \[6pt]1, & x > 10 \end{cases}
• Total probability: the area under the PDF over the entire real line equals 1:
  $\int_{-\infty}^{\infty} f(x) \, dx = 1.$
• Probability on an interval: for 4 ≤ X ≤ 8,
  $P(4 \le X \le 8) = \frac{8-4}{10} = 0.4.$
• Important for continuous variables: P(X = x) = 0 for any specific x, since the area of a single point is zero.
• Connection to the general idea: probability is the area under the density curve over the interval of interest.

Continuous vs discrete probabilities

• Key distinction: for a continuous random variable, the probability at a single point is zero:
  $P(X = x) = 0.$
• Probabilities are obtained by integrating the density over an interval (area under the curve):
  $P(a \le X \le b) = \int_{a}^{b} f(x) \, dx.$
• The total area under any PDF across the real line equals 1:
  $\int_{-\infty}^{\infty} f(x) \, dx = 1.$
• Exponential distribution appears in the discussion but not elaborated here; the same area-under-curve idea applies to all PDFs.
• For normal distributions, area under the curve between bounds corresponds to probabilities; the concept is universal for PDFs.

Normal distribution and standard normal

• General normal PDF with mean $\mu$ and standard deviation $\sigma$:
  $f(x) = \frac{1}{\sqrt{2\pi}\,\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).$
• Standard normal (special case): $Z \sim N(0,1)$ with PDF
  $\phi(z) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{z^2}{2}\right).$
• Standardization (z-transformation):
  $Z = \frac{X-\mu}{\sigma},\quad Z\sim N(0,1).$
• Cumulative distribution function (CDF) for standard normal:
  $\Phi(z) = P(Z \le z).$
• Key empirical rule (68-95-99.7):
  • $P(\mu-\sigma \le X \le \mu+\sigma) \approx 0.6827$
  • $P(\mu-2\sigma \le X \le \mu+2\sigma) \approx 0.9545$
  • $P(\mu-3\sigma \le X \le \mu+3\sigma) \approx 0.9973$
• Interpretation: these percentages describe how data from a normal distribution typically spread around the mean when using standard deviations as units.
• Relationship to standard normal table (z-table): the table gives values of (\Phi(z)), the left-tail probabilities for z.
• Practical usage: to compute probabilities for a nonstandard normal, transform to z and use the z-table or software.

Reading the standard normal table (z-table)

• What the table shows:
  • Each row corresponds to a z-value to one decimal place (e.g., -1.2, -1.3, …).
  • Each column corresponds to the second decimal place (e.g., .00, .01, .02, …).
  • The body (inside the table) gives the cumulative probability from the far left up to z, i.e., (\Phi(z)).
• How to read a value:
  • Example: z = 1.23. Find row 1.2 and column .03; the table entry is Φ(1.23).
  • For negative z, use symmetry: (\Phi(-z) = 1 - \Phi(z)).
• Example probabilities (approximate):
  • (\Phi(-1.23) \approx 0.1093)
  • (\Phi(1.23) \approx 0.8907)
  • (\Phi(1.49) \approx 0.9319)
• Using the table for a range:
  • For P(a < Z < b) = Φ(b) - Φ(a).
  • Example: P(-1.23 < Z < 2.05) = Φ(2.05) - Φ(-1.23) ≈ 0.9798 - 0.1093 ≈ 0.8705.
• Using complements:
  • P(Z > c) = 1 - Φ(c).
  • P(Z > -c) = Φ(c).

Worked examples and problem-solving steps

• Problem: Proportion of values between two z-scores for Z ~ N(0,1):
  • Given z1 = -1.23 and z2 = 2.05,
  • Probability is: $P(-1.23 \le Z \le 2.05) = \Phi(2.05) - \Phi(-1.23) \approx 0.9798 - 0.1093 \approx 0.8705.$
• Problem: Probability Z > -1.23 (right of -1.23):
  • Use complement or symmetry: P(Z > -1.23) = 1 - \Phi(-1.23) = \Phi(1.23) \approx 0.8907.
• Problem: Nonstandard normal X ~ N(\mu, \sigma^2); find P(X < x).
  • Transform to Z: Z = \frac{X-\mu}{\sigma} \implies P(X < x) = P\left(Z < \frac{x-\mu}{\sigma}\right) = \Phi\left(\frac{x-\mu}{\sigma}\right).
• Worked example: weights of packages
  • Given X ~ N(\mu=10\,\text{lb}, \sigma=4\,\text{lb}); find P(X < 14.2).
  • Step 1: compute z-score: $z = \frac{14.2-10}{4} = 1.05.$
  • Step 2: probability: P(X < 14.2) = P(Z < 1.05) = \Phi(1.05) \approx 0.8531.
• In practice, software aids:
  • R: pnorm, qnorm, rnorm
  • pnorm(q, mean, sd) gives (\Phi) for the corresponding parameters.
  • qnorm(p, mean, sd) gives the inverse CDF (the z/quantile for probability p).
  • rnorm(n, mean, sd) generates random samples from N(mean, sd^2).
  • Other software (e.g., Minitab) provides similar capabilities for computing areas and quantiles.
• Practical note: In education, recognizing the pdf and its components, and knowing how to combine with z-scores and the CDF, is often more important than memorizing calculus proofs.

PDF, CDF, and interpretation details

• Normal PDF components explained:
  • Denominator: (\sqrt{2\pi}\,\sigma) scales the width; (\sqrt{2\pi}) is a normalization constant.
  • Exponential term: $\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$ determines the bell shape centered at (\mu).
  • The term ( (x-\mu)^2 ) shows the squared distance from the mean.
  • The parameter (\mu) shifts the center; the parameter (\sigma) stretches/compresses the width.
• For a standard normal, the PDF is:
  $\phi(z) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{z^2}{2}\right).$
• The fundamental property: the area under the entire pdf curve equals 1 (total probability):
  $\int_{-\infty}^{\infty} f(x) \, dx = 1.$
• The idea of probability as area under the curve applies to all PDFs, not only the normal distribution.

Key takeaways for exam-ready understanding

• Distinguish discrete vs. continuous probabilities: only intervals have positive probability for continuous vars.
• For any PDF, total probability is 1 and P(a ≤ X ≤ b) is the area under f(x) between a and b.
• The normal distribution is fully characterized by (μ, σ):
  • Centered at μ; spread by σ.
  • Standardization enables using the standard normal table or software.
• Z-scores measure how many standard deviations an observation is from the mean:
  $Z = \frac{X-\mu}{\sigma}.$
• Interpretations using the 68-95-99.7 rule help quick assessments, but exact probabilities use Φ(z).
• When given nonstandard normal parameters, convert to Z and use Φ to find probabilities.
• Be comfortable with reading and using z-tables, and know how to compute probabilities via left-tail, right-tail, and interval probabilities through Φ values and complements.
• Familiarize yourself with common software commands for probability calculations (pnorm, qnorm, rnorm in R) to quickly obtain probabilities, quantiles, and random samples.