Chapter 5: Continuous Random Variables

These flashcards cover key vocabulary and concepts related to continuous random variables presented in the lecture notes.

Continuous Probability Density Function (pdf)

A function that represents the probability of a continuous random variable; the area under the curve represents probability.

Cumulative Distribution Function (cdf)

A function that gives the probability that a random variable takes a value less than or equal to a certain point.

Uniform Distribution

A type of continuous probability distribution where all outcomes are equally likely.

Exponential Distribution

A probability distribution that models the time until an event occurs, characterized by a constant rate of occurrence.

Probability Density Function notation

Denoted as f(x), it indicates the function used to calculate probabilities for continuous random variables.

P(c < x < d)

The probability that the random variable X falls within the interval between c and d, represented by the area under the curve.

Theoretical Mean (μ) for Uniform Distribution

Calculated using the formula (a + b) / 2, where a is the minimum value and b is the maximum value.

Standard Deviation (σ) of Uniform Distribution

Calculated using the formula (b - a)² / 12, where a is the minimum value and b is the maximum value.

Probability in Continuous Distributions

Probability is represented by the area under the curve and is calculated for intervals rather than individual points.

P(x = c) in Continuous Distributions

The probability of a continuous random variable taking on an exact value c is zero.

K in the context of percentiles

In percentile problems, k represents the threshold value such that a specified percentage of data falls below it.

Decay Parameter (m) in Exponential Distribution

The inverse of the mean (μ); m = 1/μ, representing the rate at which the probability density function decays.

P(x < k) for Exponential Distribution

Represents the cumulative distribution function, giving the probability of the random variable being less than k.

Half of all customers finished time in Exponential Distribution

The time at which 50% of customers are serviced, found using the cumulative distribution function.