Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a<b), where (a ≤ x ≤ b) and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

The probability density function of a uniform distribution is



on the interval from (x=a) to (x=b). For any value of x less than a or greater than b, y=0 . In Exercises 59 and 60, use this information.


For two values c and d, where a ≤ c < d ≤ b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below.





So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from (a=1) to (b=25) , find the probability that


d. x lies between 8 and 14.

Verified step by step guidance

1. Step 1: Understand the uniform distribution. A uniform distribution is a continuous probability distribution where all values between two bounds (a and b) are equally likely. The probability density function is constant and given by y = 1 / (b - a).
2. Step 2: Identify the bounds of the distribution. In this problem, the uniform distribution is defined between a = 1 and b = 25. The height of the probability density function is y = 1 / (b - a), which simplifies to y = 1 / (25 - 1) = 1 / 24.
3. Step 3: Recognize that the probability of x lying between two values (c and d) is equal to the area under the curve between c and d. This area is a rectangle with height y = 1 / 24 and width equal to the difference between c and d.
4. Step 4: Determine the bounds for c and d. In this case, c = 8 and d = 14. The width of the rectangle is d - c = 14 - 8 = 6.
5. Step 5: Calculate the area of the rectangle, which represents the probability. The area is given by width × height = (d - c) × (1 / (b - a)). Substitute the values: Area = (14 - 8) × (1 / 24). This area represents the probability that x lies between 8 and 14.

Key Concepts

Uniform Distribution

A uniform distribution is a type of probability distribution where all outcomes are equally likely within a specified range. For a continuous uniform distribution defined between two values, a and b, the probability density function is constant, meaning that the likelihood of the random variable falling anywhere between a and b is the same. This results in a rectangular shape when graphed, with the height determined by the formula 1/(b-a).

Probability Density Function (PDF)

The probability density function (PDF) of a continuous random variable describes the likelihood of the variable taking on a particular value. For a uniform distribution, the PDF is defined as y = 1/(b-a) for values between a and b, and y = 0 outside this interval. The area under the PDF curve between any two points represents the probability that the random variable falls within that range.

Area Under the Curve

In the context of probability distributions, the area under the curve of the PDF represents the probability of the random variable falling within a specific interval. For a uniform distribution, this area can be calculated as the product of the width of the interval (d-c) and the height of the rectangle (1/(b-a)). This concept is crucial for determining probabilities in continuous distributions.