Section 4.5–4.6: Continuous Random Variables & the Normal Distribution

Continuous Random Variables & Probability Density Functions (PDFs)

  • Review from previous section 4.4

    • Last week focused on discrete random variables (probability mass functions, sums of probabilities).

    • Key bridge: many discrete models (e.g.

    • Binomial) can be approximated by continuous models (especially the normal) for large sample sizes.

  • Fundamental distinction

    • Continuous random variable X takes on every value in an interval of the real line.

    • Because values are uncountably infinite, P(X = x) = 0 for any single point; probabilities are only meaningful over intervals.

  • Probability Density Function (PDF) f(x)

    • Smooth curve drawn over the real-number axis; sometimes called frequency function or just probability distribution.

    • Area under the curve over an interval corresponds to probability:
      P(a \le X \le b) = \int_a^b f(x)\,dx

    • Total area under entire curve is 1.

    • Graphical intuition: shade region from a to b → shaded area gives the desired probability.

The Normal Distribution $\mathcal N(\mu,\sigma^2)$

  • Central role

    • Describes numerous natural/industrial phenomena (heights, measurement error, IQ, bulb lifetime, etc.).

    • Serves as a limiting approximation for many discrete distributions (notably binomial via De Moivre–Laplace/Central Limit Theorem).

    • Forms bedrock of classical statistical inference (confidence intervals, hypothesis testing).

  • Shape & descriptive characteristics

    • Bell-shaped, symmetrical (a.k.a. mound-shaped).

    • Mean, median, and mode coincide at x = \mu (curve is perfectly balanced there).

    • Inflection points located one standard deviation on either side (at \mu \pm \sigma).

  • PDF formula f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{- \frac12\bigl(\frac{x-\mu}{\sigma}\bigr)^2}

    • Parameters:

    • \mu = mean (center)

    • \sigma = standard deviation (spread)

    • Transcendental constants e \approx 2.71828 and \pi \approx 3.14159 appear.

    • Memorization not required; recognition and parameter interpretation are essential.

  • Visual parameter effects (3 example curves)

    1. \mu = 0,\; \sigma = 1 → standard curve (purple), widest of shown.

    2. \mu = 3,\; \sigma = 1 → curve shifted right, same spread.

    3. \mu = -4,\; \sigma = 0.5 → curve shifted left, narrower and taller because smaller \sigma.

  • Area–probability equivalence holds: P(c \le X \le d) = \int_c^d f(x)\,dx.

    • If calculus unfamiliar, mentally replace integral by area of shaded strip.

The Standard Normal Distribution $Z \sim \mathcal N(0,1)$

  • Special case with \mu = 0,\; \sigma = 1.

  • Variable usually denoted Z.

  • Historically, probabilities obtained from printed $Z$-tables:

    • E.g. P(0 \le Z \le 1.96) = 0.4750 from table (row 1.9, column 0.06).

  • Modern workflow: use calculators or software:

    • TI-84: 2nd → DIST (VARS) → normalcdf(lower, upper).

    • Example: normalcdf(-1.26,1.26) returns 0.7923; table gives slightly less accurate 0.7922 (rounding).

Standardising a General Normal Variable

  • Any normal X \sim \mathcal N(\mu,\sigma^2) can be converted to Z via Z = \frac{X-\mu}{\sigma}

    • Called z-score; measures distance from mean in units of standard deviation.

    • Enables use of standard normal tables or functions.

  • Procedure for probability questions

    1. Sketch bell curve; mark mean and shade desired area.

    2. Convert interval bounds x1, x2 to z-scores:
      z1 = \frac{x1-\mu}{\sigma},\quad z2 = \frac{x2-\mu}{\sigma}

    3. Use normalcdf(z_1,z_2).

    4. Translate result back if interpreting in X context.

  • Calculator shortcut: TI-84 allows normalcdf(lower,upper,μ,σ) directly, but understanding standardisation remains conceptually critical.

Worked Forward-Probability Examples

  1. X \sim \mathcal N(5,10^2)

    • Find P(5 \le X \le 6.2).

      • z-scores: z1 = 0 (5 at mean), z2 = (6.2-5)/10 = 0.12.

      • normalcdf(0,0.12) → 0.0478.

  2. Same distribution, P(2.9 \le X \le 7.1).

    • z1 = (2.9-5)/10 = -0.21, z2 = +0.21.

    • normalcdf(-0.21,0.21) → 0.1663.

  3. Light-bulb lifetime

    • \mu = 2000 h, \sigma = 200 h (X \sim \mathcal N(2000,200^2)).

    • a) P(2000 \le X \le 2400).

      • z1 = 0, z2 = (2400-2000)/200 = 2.

      • Area: normalcdf(0,2) → 0.4772.

    • b) P(X < 1470).

      • z = (1470-2000)/200 = -2.65.

      • Two approaches:

      1. Compute left-tail directly: normalcdf(-1e99,-2.65) → 0.0040.

      2. Compute strip to 0 then subtract from 0.5: normalcdf(-2.65,0) = 0.4960; 0.5 - 0.4960 = 0.0040.

    • Interpretation: Very unlikely (0.4 %) that a bulb fails before 1470 h.

Inverse Normal Problems (Finding Quantiles)

  • Given a probability (area), find z such that P(Z \le z) = p.

  • TI-84: invNorm(p) (or inverse normal in menu).

    • Input = area to left of desired point.

    • Output = quantile (z-value).

Examples

  1. Determine a such that P(0 \le Z \le a) = 0.1217.

    • Total area to left of a: 0.5 + 0.1217 = 0.6217.

    • invNorm(0.6217) → a \approx 0.3099 \approx 0.31.

  2. Find z0 with P(Z \le z0) = 0.2858.

    • Left area smaller than 0.5 ⇒ quantile is negative.

    • invNorm(0.2858) → z_0 \approx -0.57.

  3. Symmetric central probability: find z0 s.t. P(-z0 \le Z \le z_0) = 0.7294.

    • Tail area combined: 1 - 0.7294 = 0.2706; each tail 0.1353.

    • Focus on left tail: invNorm(0.1353) → -1.1.

    • Therefore z_0 = 1.1.

Inverse for Non-standard Normals

  • If P(a \le X \le b) is given and X \sim \mathcal N(\mu,\sigma^2):

    1. Convert problem to Z via standardisation.

    2. Use invNorm to get appropriate z-quantile(s).

    3. Convert back: x = \mu + z\sigma.

Worked Example

  • X \sim \mathcal N(5,10^2); find a s.t. P(5 \le X \le a) = 0.1217.

    1. Diagram: strip to right of mean with given area.

    2. Corresponding Z problem identical to earlier 0 \le Z \le z with area 0.1217 → z = 0.3099.

    3. Translate: a = \mu + z\sigma = 5 + 0.3099(10) \approx 8.1.

Practical, Philosophical & Cross-Lecture Connections

  • Practical relevance: Engineers, scientists, and quality-control analysts rely on normal probabilities to predict defect rates (e.g., bulb life example) and design tolerance intervals.

  • Ethical dimension: Misinterpreting probabilities (e.g., ignoring tail risk) can lead to faulty decision-making in public policy or product safety.

  • Link to earlier material: The binomial distribution—treated in the discrete section—converges to the normal when number of trials is large and p not extreme (De Moivre-Laplace). Understanding standardisation prepares you for the Central Limit Theorem in later chapters.

  • Tool mindset: While technology replaces hand-integration, conceptual mastery (areas ↔ probabilities, standardisation, tails) is vital for correct model choice and critical evaluation of software outputs.

Summary Checklist

  • [ ] Recognise definition and properties of a PDF for continuous variables.

  • [ ] Memorise that probabilities = areas under the curve.

  • [ ] Identify & interpret \mu and \sigma in any normal PDF.

  • [ ] Convert X \sim \mathcal N(\mu,\sigma^2) ↔ Z \sim \mathcal N(0,1) via Z = (X-\mu)/\sigma.

  • [ ] Use normalcdf for forward probabilities, invNorm for quantiles.

  • [ ] Always sketch curve, mark means, endpoints, and shade area—this prevents sign mistakes & mis-ordering of bounds.