📈 Continuous Probability

1. Continuous Random Variables

Unlike discrete random variables that take specific countable values, a continuous random variable can take any value within a range or interval. These variables are typically measurements rather than counts.

Discrete Random Variables

• Countable values

• Can list all possibilities

• Examples: number of students, dice rolls, coin flips

• P(X = specific value) > 0

Continuous Random Variables

• Uncountable, infinite values

• Values in a range

• Examples: height, weight, time, temperature

• P(X = specific value) = 0

Critical Concept:
For continuous random variables, the probability of any exact value is zero! P(X = 5.0000...) = 0

Instead, we find probabilities for intervals:
• P(a < X < b) = probability X is between a and b
• P(X < a) = probability X is less than a
Example 1: Why Exact Values Have Zero Probability

Consider measuring someone's height. Is their height exactly 170 cm?

• Not 170.1 cm or 169.9 cm

• But exactly 170.000000... cm (infinite precision)?

• The probability is essentially 0

Instead, we ask: P(169.5 < height < 170.5) = meaningful probability

📝 Practice Question

Which of the following is a continuous random variable?
A) Number of customers in a store
B) Time until a lightbulb burns out
C) Number of defects in a product
D) Number of heads in 10 coin flips

2. Probability Density Functions (PDF)

A probability density function (PDF) describes the distribution of a continuous random variable. The PDF doesn't give probabilities directly, but the area under the curve represents probability.

For a PDF f(x):

• f(x) ≥ 0 for all x
• Total area under the curve = 1
• P(a < X < b) = area under f(x) from a to b
Probability as Area Under the Curve
μ a b P(a < X < b)

The shaded area represents the probability that X falls between a and b

Important Properties:
• The PDF curve itself is NOT a probability (can be > 1)
• Only the AREA under the curve gives probability
• Total area under entire curve always equals 1
• Probabilities are found using integration (calculus)

📝 Practice Question

For a continuous random variable, what represents probability?
A) The height of the PDF curve
B) The area under the PDF curve
C) The slope of the PDF curve
D) The maximum value of the PDF

3. The Normal Distribution

The normal distribution (also called the Gaussian distribution or bell curve) is the most important continuous probability distribution. It appears everywhere in nature and statistics!

Normal Distribution: X ~ N(μ, σ²)

Parameters:
• μ (mu) = mean (center of distribution)
• σ (sigma) = standard deviation (spread)
• σ² = variance
The Normal Distribution Curve
μ μ-σ μ+σ
Properties of Normal Distribution:
• Symmetric bell-shaped curve
• Mean = Median = Mode (all at the center)
• Determined by μ and σ
• Tails extend infinitely (but probability diminishes)
• Area under entire curve = 1
Example 2: Real-World Normal Distributions

Common examples of normally distributed data:

• Heights of adult males: μ ≈ 70 inches, σ ≈ 3 inches

• IQ scores: μ = 100, σ = 15

• Blood pressure: μ ≈ 120 mmHg, σ ≈ 10 mmHg

• Test scores (when well-designed): μ = class average, σ varies

• Measurement errors in scientific experiments

📝 Practice Question

In a normal distribution, what percentage of the curve is on each side of the mean?
A) 68%
B) 50%
C) 95%
D) 100%

4. The Empirical Rule (68-95-99.7 Rule)

The Empirical Rule tells us what percentage of data falls within certain distances from the mean in a normal distribution. This is one of the most useful rules in statistics!

The 68-95-99.7 Rule

68% of data falls within 1 standard deviation of the mean
(between μ - σ and μ + σ)
95% of data falls within 2 standard deviations of the mean
(between μ - 2σ and μ + 2σ)
99.7% of data falls within 3 standard deviations of the mean
(between μ - 3σ and μ + 3σ)
Example 3: IQ Scores

IQ scores are normally distributed with μ = 100 and σ = 15.

Using the Empirical Rule:

• 68% of people have IQs between 85 and 115 (100 ± 15)
• 95% of people have IQs between 70 and 130 (100 ± 30)
• 99.7% of people have IQs between 55 and 145 (100 ± 45)
Only about 0.15% have IQs above 145 or below 55!

📝 Practice Question

Heights of women are normally distributed with mean 64 inches and standard deviation 2.5 inches. Approximately what percentage of women are between 59 and 69 inches tall?
A) 68%
B) 95%
C) 99.7%
D) 50%

5. Standard Normal Distribution and Z-Scores

The standard normal distribution is a special normal distribution with μ = 0 and σ = 1. We can convert any normal distribution to the standard normal using z-scores.

Z-Score Formula:

z = (x - μ) / σ

A z-score tells us how many standard deviations a value is from the mean
Interpreting Z-Scores:
• z = 0: value equals the mean
• z = 1: value is 1 standard deviation above the mean
• z = -1: value is 1 standard deviation below the mean
• z = 2.5: value is 2.5 standard deviations above the mean
• |z| > 3: unusual/rare value (outside 99.7% of data)
Example 4: Calculating Z-Scores

Test scores: μ = 75, σ = 10. Find z-scores for the following:

Solutions:

Score of 85: z = (85 - 75) / 10 = 1.0 (1 SD above mean)
Score of 60: z = (60 - 75) / 10 = -1.5 (1.5 SD below mean)
Score of 75: z = (75 - 75) / 10 = 0 (exactly at mean)
Score of 100: z = (100 - 75) / 10 = 2.5 (2.5 SD above mean - very high!)
Partial Standard Normal (Z) Table

Values show the area to the LEFT of z

z 0.00 0.01 0.02 0.03
0.0 0.5000 0.5040 0.5080 0.5120
1.0 0.8413 0.8438 0.8461 0.8485
2.0 0.9772 0.9778 0.9783 0.9788

Example: P(Z < 2.0) = 0.9772 or 97.72%

📝 Practice Question

A student scores 88 on a test where the mean is 75 and standard deviation is 5. What is their z-score?
A) 1.3
B) 2.0
C) 2.6
D) 13

6. Real-World Applications

Example 5: Quality Control

A factory produces bolts with diameter normally distributed: μ = 10 mm, σ = 0.2 mm. Bolts are rejected if diameter is outside 9.5 to 10.5 mm.

What percentage are rejected?

Range 9.5 to 10.5 is μ ± 2.5σ
Within 2 SD: 95% are acceptable
Within 3 SD: 99.7% are acceptable
2.5σ is between these, approximately 98.8% acceptable
About 1.2% are rejected

📝 Practice Question

Why is the normal distribution so important in statistics and real-world applications?
A) It's the easiest distribution to calculate
B) It only applies to human characteristics
C) Many natural phenomena follow it, and sample means tend toward it
D) It's the only continuous distribution

🎯 Lesson Summary

Fantastic work! You've mastered continuous probability. Here's what you learned:

  • ✅ Continuous random variables: Take any value in a range; P(exact value) = 0
  • ✅ PDF: Probability density function; area under curve = probability
  • ✅ Normal distribution: Bell curve defined by μ and σ
  • ✅ Empirical Rule: 68-95-99.7 for data within 1-2-3 standard deviations
  • ✅ Z-scores: z = (x - μ) / σ; standardize any normal distribution
  • ✅ Applications: Heights, test scores, quality control, and much more!

Next lesson: We'll explore advanced probability topics to complete your journey!