If Continuous Random Variable X Follow Distribution and Calculate

When a continuous random variable X follows a specific probability distribution, we can calculate various statistical properties such as probabilities, expected values, and variances. This guide explains how to perform these calculations for common continuous distributions.

Common Continuous Distributions

Several continuous probability distributions are commonly used in statistics and probability theory. The most important ones include:

Normal Distribution - Symmetric bell-shaped distribution characterized by mean μ and standard deviation σ
Uniform Distribution - All outcomes are equally likely within a specified range [a, b]
Exponential Distribution - Models the time between events in a Poisson process with rate λ
Gamma Distribution - Generalization of the exponential distribution with shape parameter k and scale θ
Beta Distribution - Used for modeling proportions and probabilities with shape parameters α and β

Probability Calculation

To calculate the probability that a continuous random variable X falls within a specific range [a, b], we use the cumulative distribution function (CDF):

P(a ≤ X ≤ b) = F(b) - F(a)

Where F(x) is the cumulative distribution function of X

For example, if X follows a standard normal distribution (μ=0, σ=1), we can use standard normal tables or statistical software to find F(b) and F(a).

Expected Value

The expected value (mean) of a continuous random variable X is calculated as:

E[X] = ∫x·f(x) dx

Where f(x) is the probability density function of X

For many common distributions, the expected value can be calculated using simpler formulas:

Normal: E[X] = μ
Uniform: E[X] = (a + b)/2
Exponential: E[X] = 1/λ

Variance

The variance of a continuous random variable X measures the spread of its distribution:

Var(X) = E[X²] - (E[X])² = ∫x²·f(x) dx - (∫x·f(x) dx)²

For common distributions, simplified formulas exist:

Normal: Var(X) = σ²
Uniform: Var(X) = (b - a)²/12
Exponential: Var(X) = 1/λ²

Practical Examples

Example 1: Normal Distribution

Suppose X follows a normal distribution with μ=50 and σ=10. Calculate P(40 ≤ X ≤ 60):

P(40 ≤ X ≤ 60) = F(60) - F(40)

Using standard normal tables:

F(60) ≈ 0.9772

F(40) ≈ 0.0228

Result: 0.9772 - 0.0228 = 0.9544 or 95.44%

Example 2: Uniform Distribution

If X is uniformly distributed between 10 and 20, calculate E[X] and Var(X):

E[X] = (10 + 20)/2 = 15

Var(X) = (20 - 10)²/12 = 100/12 ≈ 8.333

Frequently Asked Questions

What is the difference between probability density function and cumulative distribution function?

The probability density function (PDF) gives the relative likelihood of X taking on a given value, while the cumulative distribution function (CDF) gives the probability that X will take a value less than or equal to x.

How do I know which distribution to use for my data?

You can use statistical tests or graphical methods to determine which distribution best fits your data. Common distributions like normal, exponential, and uniform often provide good fits for many real-world phenomena.

What is the relationship between variance and standard deviation?

The standard deviation is the square root of the variance. While variance measures the spread in squared units, standard deviation is in the same units as the original data.