Chapter 6

Chapter 6: Continuous Probability Distributions

(Phân phối Xác suất Liên tục)Dr. Pham Phuong Ngoc

This chapter introduces continuous probability distributions, explains how probabilities are determined as areas under a curve, and focuses on the normal and standard normal distributions—two of the most important continuous distributions in statistics.

I. Continuous Probability Distribution

(Phân phối Xác suất Liên tục)

Definition:

A continuous random variable (Biến ngẫu nhiên liên tục) can assume any value within an interval (or a collection of intervals) on the real number line. Unlike discrete variables, the probability of the variable taking on any one exact value is zero.(Note: P(X = x) = 0 for any specific value x; only intervals have nonzero probabilities.)

How It Works:

Since assigning probability to a single point is impossible, we find the probability that the variable falls within an interval [x1, x2]. This probability is determined by the area under the probability density function (PDF) between x1 and x2.

II. Uniform Probability Distribution

(Phân phối Đồng đều)

Uniform Distribution:

A random variable is uniformly distributed (phân phối đồng đều) if every interval of equal length within its range has the same probability.

Probability Density Function (PDF) for Uniform Distribution on [a, b]:

• f(x) = 1/(b - a) for a ≤ x ≤ b

• f(x) = 0 elsewhere

Expected Value and Variance:

• Expected Value, E(X) = (a + b) / 2

• Variance, Var(X) = (b - a)² / 12

Example:

If X is uniformly distributed between 10 and 20, then:

• P(X < 15) is proportional to the length of the interval from 10 to 15, and

• E(X) = (10 + 20)/2 = 15.

III. Normal Probability Distribution

(Phân phối Chuẩn / Phân phối Chuẩn thường)

Definition:

The normal distribution (phân phối chuẩn) is a continuous distribution that is symmetrical, bell-shaped, and defined by two parameters: the mean (mu) and the standard deviation (sigma).

Probability Density Function (PDF) for the Normal Distribution:

f(x) = 1 / (sigma √(2π)) exp(- (1/2) ((x - mu) / sigma)²)

Characteristics:

• The curve is symmetric about mu, meaning the mean, median, and mode are all equal.

• The standard deviation (sigma) determines the spread of the curve; a larger sigma results in a wider, flatter curve.

• The total area under the curve equals 1.

Empirical (68–95–99.7) Rule:

• Approximately 68.3% of values lie within (mu - sigma) to (mu + sigma)

• Approximately 95.4% lie within (mu - 2sigma) to (mu + 2sigma)

• Approximately 99.7% lie within (mu - 3sigma) to (mu + 3sigma)

IV. Standard Normal Probability Distribution

(Phân phối Chuẩn Chuẩn hóa)

Definition:

The standard normal distribution (phân phối chuẩn chuẩn hóa) is a special case of the normal distribution with a mean of 0 and a standard deviation of 1.

Standardization:

Any normal random variable X ~ N(mu, sigma²) can be converted into a standard normal variable Z using:z = (x - mu) / sigma

PDF for the Standard Normal Distribution:

f(z) = 1 / √(2π) * exp(- z² / 2)

Probability Calculations Using Z:

• P(Z < b) = Phi(b)

• P(Z > a) = 1 - Phi(a)

• P(a < Z < b) = Phi(b) - Phi(a)(Use the standard normal table or software to find these values.)

Example:

For X ~ N(400, 10²), to compute P(400 < X < 415), first standardize by calculating z = (X - 400)/10 and then use the standard normal table to determine the probability.

Key Terms and Translations:

• Continuous Random Variable: Biến ngẫu nhiên liên tục

• Probability Density Function (PDF): Hàm mật độ xác suất

• Uniform Distribution: Phân phối đồng đều

• Normal Distribution: Phân phối chuẩn

• Standard Normal Distribution: Phân phối chuẩn chuẩn hóa

• Expected Value: Giá trị kỳ vọng

• Standard Deviation: Độ lệch chuẩn

• Empirical Rule: Quy tắc 68–95–99.7