Applications of Uniform Distribution and Probability Density Functions in Everyday Scenarios

Introduction:

Review the practical applications of uniform distributions and probability density functions (PDFs) in real-world scenarios. Through an example of randomly selecting a number from the interval [2,10] shows how these mathematical concepts are utilized in various fields, such as quality control, simulations, financial risk modeling, and service management. The example provides insights into calculating probabilities for specific events, such as exceeding thresholds, or falling within a specified range, using uniform distribution.

Probability density functions (PDFs) are fundamental to understanding randomness in a continuous range. Uniform distribution, a type of probability distribution where all outcomes are equally likely, is commonly encountered in everyday situations. This paper aims to explain the use of uniform distributions and PDFs using an example where a real number XXX is randomly selected from the interval [2,10][2, 10][2,10]. The methodology employed here can be directly applied to a range of real-world scenarios.

Problem Statement:

Given a real number XXX chosen uniformly from the interval [2,10][2, 10][2,10], we aim to:

1. Find the probability density function f(x)f(x)f(x).
2. Calculate the probability for specific events X>5X > 5X>5, 5<X<75 < X < 75<X<7, and X2−12X+35>0X^2 – 12X + 35 > 0X2−12X+35>0.

Solution:

(a) Finding the Density Function f(x)f(x)f(x):

For a uniform distribution over the interval [2,10][2,10][2,10], the probability density function is constant and can be defined as:f(x)=110−2=18forx∈[2,10]f(x) = \frac{1}{10 – 2} = \frac{1}{8} \quad \text{for} \quad x \in [2,10]f(x)=10−21​=81​forx∈[2,10]

This implies that the likelihood of selecting any number within the interval is evenly distributed.

(b) Calculating Probabilities for Events:

Using the density function derived above, we now compute the probability of specific events.

1. Probability that X>5X > 5X>5:

P(X>5)=10−58=58P(X > 5) = \frac{10 – 5}{8} = \frac{5}{8}P(X>5)=810−5​=85​

2. Probability that 5<X<75 < X < 75<X<7:

P(5<X<7)=7−58=28=14P(5 < X < 7) = \frac{7 – 5}{8} = \frac{2}{8} = \frac{1}{4}P(5<X<7)=87−5​=82​=41​

3. Probability that X2−12X+35>0X^2 – 12X + 35 > 0X2−12X+35>0: Factoring the quadratic expression X2−12X+35>0X^2 – 12X + 35 > 0X2−12X+35>0 gives:

(X−5)(X−7)>0(X – 5)(X – 7) > 0(X−5)(X−7)>0

This inequality holds when X<5X < 5X<5 or X>7X > 7X>7. Therefore, the probability is:P(X2−12X+35>0)=P(X<5 or X>7)=P([2,5]∪[7,10])=38+38=68=34P(X^2 – 12X + 35 > 0) = P(X < 5 \, \text{or} \, X > 7) = P([2,5] \cup [7,10]) = \frac{3}{8} + \frac{3}{8} = \frac{6}{8} = \frac{3}{4}P(X2−12X+35>0)=P(X<5orX>7)=P([2,5]∪[7,10])=83​+83​=86​=43​

Applications in Real Life:

1. Quality Control in Manufacturing: In quality testing, products may have a uniform lifespan distribution between a set interval (e.g., 2 to 10 hours). The probability of a product failing after a certain number of hours is calculated using a uniform distribution.
2. Simulation of Random User Behavior: When simulating how long users stay on a webpage (within a time frame of 2 to 10 minutes), a uniform distribution can help predict the likelihood that users will remain on the page for a given period.
3. Financial Risk Assessment: A financial portfolio may have uniform expected returns in a set range (e.g., 2% to 10%). The uniform distribution helps calculate the probability of achieving a return greater than a specific threshold, such as 5%.

Code Example:

The following Python code can be used to calculate the probabilities described:

def probability_uniform_interval(a, b, lower_bound, upper_bound):
    total_interval = b - a
    probability = (upper_bound - lower_bound) / total_interval
    return probability

# Example calculations
P_X_greater_than_5 = probability_uniform_interval(2, 10, 5, 10)
P_5_less_than_X_less_than_7 = probability_uniform_interval(2, 10, 5, 7)

# For quadratic probability X^2 - 12X + 35 > 0
P_X_squared_condition = probability_uniform_interval(2, 10, 2, 5) + probability_uniform_interval(2, 10, 7, 10)

print("P(X > 5):", P_X_greater_than_5)
print("P(5 < X < 7):", P_5_less_than_X_less_than_7)
print("P(X^2 - 12X + 35 > 0):", P_X_squared_condition)

Conclusion:

The uniform distribution and probability density functions are widely used in various real-world contexts, including quality control, simulations, and risk modeling. The principles of continuous probability help calculate the likelihood of outcomes over an interval, enabling informed decision-making in diverse fields.

References:

Introduction to Probability, second edition (Chapman & Hall/CRC Texts in Statistical Science) (Hardcover) | San Marco Books and more. (n.d.). https://www.sanmarcobooksandmore.com/book/9781138369917