CLASS 8

Chapter 5: Normal Distribution (Part I)

Normal Distribution

• Definition: A continuous random variable with a symmetric, bell-shaped distribution centered at the mean ($\mu$).

  • Probability Density Function (PDF): $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$

  • $\mu$ = population mean

  • $\sigma$ = population standard deviation

Properties of Normal Distribution Curve:

• Total Area: Equals $1$.

• Bell-Shaped: Characteristic curve shape.

• Symmetric: Around the mean ($\mu$).

• Centering: At the mean ($\mu$).

• Asymptotic: Tails approach, but never touch, the x-axis.

Graph Characteristics:

• Determined by $\mu$ and $\sigma$.

• Change in $\sigma$: Wider (larger $\sigma$) or narrower (smaller $\sigma$).

• Change in $\mu$: Shifts graph left or right.

Empirical Rule for Bell-Shaped Distributions (68-95-99.7)

• Valid for approximately normal distributions:

  • 68% Rule: Approximately $68\%$ of values lie within $\pm 1$ standard deviation of the mean $[\mu - \sigma, \mu + \sigma]$.

  • 95% Rule: Approximately $95\%$ of values fall within $\pm 2$ standard deviations of the mean $[\mu - 2\sigma, \mu + 2\sigma]$. (Usual values)

  • 99.7% Rule: Approximately $99.7\%$ of values are within $\pm 3$ standard deviations of the mean $[\mu - 3\sigma, \mu + 3\sigma]$. (Almost all values)

Standard Normal Distribution

• Definition: A special normal distribution with $\mu = 0$ and $\sigma = 1$, denoted as $Z \sim N(0, 1)$.

• Characteristics: Always centered at $0$, horizontal axis values are z-scores.

• Z-Score Formula: $Z = \frac{X - \mu}{\sigma}$

  • Positive z-score: above the mean.

  • Negative z-score: below the mean.

  • Z-score of $0$: at the mean.

• Probability: Area under the density curve corresponds to probabilities.

Types of Problems:

1. Given z-score: Find probability (area).

2. Given probability (area): Find z-score.

Case 1: Finding Probability Using z-Scores

• Example 1 (P(z < 1)): For $z = 1$, area from z-table is $0.8413$. So, P(z < 1) = 0.8413.

• Example 2 (P(z > -1.02)): P(z > -1.02) = 1 - P(z < -1.02) = 1 - 0.1539 = 0.8461.

Case 2: Finding z-Score Given Probability

• Example ($85^{th}$ percentile): For an area of $0.85$, the z-score is approximately $1.036$ (looking up $0.8500$ in the z-table).

Example 4: Calculate Probabilities for $N(0, 1)$

1. P(Z > 3.58) = 1 - P(Z < 3.58) = 1 - 0.9998 = 0.0002.

2. P(Z > -1.73) = 1 - P(Z < -1.73) = 1 - 0.0418 = 0.9582.

3. P(1.24 < Z < 2.58) = P(Z < 2.58) - P(Z < 1.24) = 0.9951 - 0.8925 = 0.1026.

Important Formulas:

• P(Z < +a) = \text{use positive table}

• P(Z < -a) = \text{use negative table}

• P(Z > a) = 1 - P(Z < a)

• P(a < Z < b) = P(Z < b) - P(Z < a)

Test Your Knowledge

1. Calculate Area (P(Z > -2.72)): 1 - P(Z < -2.72) = 1 - 0.0033 = 0.9967.

2. Determine z-score (for $0.12$ area): Approximately $z = -1.175$.

3. Find Probabilities:

   • P(Z < 1.72) = 0.9573

   • P(Z < -2.10) = 0.0179

   • P(Z < 0.89) = 0.8133

4. Calculate Probabilities:

   • P(Z > 1) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587

   • P(Z > 2.54) = 1 - P(Z < 2.54) = 1 - 0.9945 = 0.0055

   • P(-0.52 < Z < 1.80) = P(Z < 1.80) - P(Z < -0.52) = 0.9641 - 0.3015 = 0.6626

5. Determine z-score for areas (Critical Values):

   • $\pm 1.645$ (for central $$90\