7.1: Continuous Probability Distributions Flashcards
What is a continuous probability distribution?
A continuous probability distribution assigns probabilities to intervals of values for a random variable that can take any numerical value within an interval on the real line.
Define a continuous random variable.
A continuous random variable is one that can assume any numerical value in one or more intervals on the real number line.
What is the significance of the area under the curve in a continuous probability distribution?
The area under the curve represents the probability that the continuous random variable will fall within a specified interval.
How is a probability density function represented graphically?
A probability density function is represented by a curve on a graph, where the area under the curve between two points corresponds to the probability of the variable falling within that interval.
What does the probability P(153 ≤ x ≤ 167) = .7223 signify in the context of coffee temperature?
It signifies that there is a 72.23% chance that the temperature of a randomly selected cup of coffee will be between 153°F and 167°F.
What do we call a curve that represents a continuous probability distribution?
It is referred to as a probability curve or a probability density function.
What are the two conditions that a continuous probability distribution must satisfy?
1) The function f(x) must be greater than or equal to 0 for any value of x.
2) The total area under the curve of f(x) must be equal to 1.
What is the probability that a continuous random variable equals a single value?
The probability that a continuous random variable equals a single numerical value is always 0.
What is meant by the probability P(a < x < b) for a continuous random variable?
It represents the probability that the variable x will fall within the interval between a and b, not including the endpoints themselves.
What is a normal probability distribution?
It is a continuous probability distribution that is symmetrical and bell-shaped, used to describe random variables that tend to cluster around a single mean value.
