Calculate The Following Probabilities Directly From The Cdf

What is a Cumulative Distribution Function (CDF)?

The cumulative distribution function (CDF) is a statistical function that provides the probability that a random variable X will take a value less than or equal to a specified value x. For a continuous random variable, the CDF is defined as:

The CDF is related to the probability density function (PDF) through integration. For a continuous random variable, the CDF can be obtained by integrating the PDF from negative infinity to x:

Where f(t) is the PDF of the random variable X. The CDF has several important properties:

• It is non-decreasing: If x1 ≤ x2, then F(x1) ≤ F(x2)
• It approaches 0 as x approaches negative infinity
• It approaches 1 as x approaches positive infinity
• It is right-continuous

The CDF provides a complete description of the probability distribution of a random variable. It can be used to calculate probabilities for various ranges of values, including:

• P(a ≤ X ≤ b) = F(b) - F(a)
• P(X > a) = 1 - F(a)
• P(X < b) = F(b)

Calculating Probabilities Directly from CDF

Calculating probabilities directly from the CDF involves using the properties of the CDF to find the probability that a random variable falls within a specific range. The key formula for calculating the probability that X falls between two values a and b is:

This formula works because the CDF gives the cumulative probability up to a certain point. By subtracting the CDF evaluated at the lower bound from the CDF evaluated at the upper bound, you get the probability that X falls within that range.

For example, if you want to find the probability that a normally distributed random variable X is between 1.5 and 2.5, you would:

1. Find the CDF of the normal distribution at 2.5
2. Find the CDF of the normal distribution at 1.5
3. Subtract the second value from the first

This approach is particularly useful when working with standard probability distributions where the CDF can be expressed in closed form or can be easily computed using statistical software or calculators.

Example Calculation

Let's consider a standard normal distribution (mean = 0, standard deviation = 1) and calculate the probability that X is between -1 and 1.

For a standard normal distribution, the CDF can be calculated using the standard normal cumulative distribution function, often denoted as Φ(x).

Using the formula:

Looking up the values in standard normal distribution tables or using a calculator:

• Φ(1) ≈ 0.8413
• Φ(-1) ≈ 0.1587

Therefore:

This means there is approximately a 68.26% probability that a standard normal random variable falls between -1 and 1.

This example demonstrates how to use the CDF to calculate probabilities for a specific range. The same approach can be applied to other probability distributions by using their respective CDF functions.

Common Distributions and Their CDFs

Several common probability distributions have well-defined CDFs that can be used to calculate probabilities directly. Here are some examples:

Normal Distribution

The CDF of the normal distribution with mean μ and standard deviation σ is given by:

Where erf is the error function. For the standard normal distribution (μ = 0, σ = 1), this simplifies to Φ(x).

Exponential Distribution

The CDF of the exponential distribution with rate parameter λ is:

This distribution is often used to model the time between events in a Poisson process.

Uniform Distribution

The CDF of the continuous uniform distribution over the interval [a, b] is:

This distribution is used when all outcomes in a range are equally likely.

Binomial Distribution

The CDF of the binomial distribution with parameters n and p is:

Where C(n, i) is the binomial coefficient. This distribution is used to model the number of successes in a fixed number of independent Bernoulli trials.

Understanding the CDFs of these common distributions allows you to calculate probabilities directly for a wide range of statistical problems.