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 $x<em>1, x</em>2$ to z-scores:
     $z<em>1 = \frac{x</em>1-\mu}{\sigma},\quad z<em>2 = \frac{x</em>2-\mu}{\sigma}$

  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: $z<em>1 = 0$ (5 at mean), $z</em>2 = (6.2-5)/10 = 0.12$.

     • normalcdf(0,0.12) → $0.0478$.

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

   • $z<em>1 = (2.9-5)/10 = -0.21$, $z</em>2 = +0.21$.

   • 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) $P(2000 \le X \le 2400)$.

     • $z<em>1 = 0$, $z</em>2 = (2400-2000)/200 = 2$.

     • 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 $z<em>0$ with $P(Z \le z</em>0) = 0.2858$.

   • Left area smaller than 0.5 ⇒ quantile is negative.

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

3. Symmetric central probability: find $z<em>0$ s.t. $P(-z</em>0 \le Z \le z_0) = 0.7294$.

   • 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.