11.1 Uniform and exponential distributions

Uniform and exponential distributions are key players in probability theory. Uniform distribution spreads probability evenly across an interval, while exponential distribution models time between events in a Poisson process.

These distributions have unique properties and applications. Uniform is great for modeling random selection, while exponential shines in scenarios like customer arrivals or component lifespans. Understanding their characteristics helps solve real-world probability problems.

Uniform Distribution

Properties of uniform distributions

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• Uniform distribution represents a continuous probability distribution where the probability density function (PDF) remains constant over a defined interval $[a, b]$
• PDF is given by $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$, and $0$ otherwise, meaning the probability is evenly distributed across the interval
• Mean of the uniform distribution is the midpoint of the interval, calculated as $\mu = \frac{a+b}{2}$
• Variance measures the spread of the distribution, given by $\sigma^2 = \frac{(b-a)^2}{12}$
• Cumulative distribution function (CDF) represents the probability that a random variable $X$ is less than or equal to a value $x$, expressed as $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
• Uniform distribution is symmetrical about the midpoint of the interval, meaning the probability of a value occurring is the same on both sides of the midpoint

Probabilities in uniform distributions

• Probability of a random variable $X$ falling within the interval $[a, b]$ is always 1, as $P(a \leq X \leq b) = \frac{b-a}{b-a} = 1$
• Probability of $X$ being less than or equal to a value $x$ within the interval is given by the CDF, $P(X \leq x) = F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
  • Example: If $X$ follows a uniform distribution over the interval $[2, 8]$, then $P(X \leq 5) = \frac{5-2}{8-2} = \frac{1}{2}$
• Probability of $X$ being greater than a value $x$ within the interval is the complement of the CDF, $P(X > x) = 1 - F(x) = 1 - \frac{x-a}{b-a}$ for $a \leq x \leq b$
• $p$-th quantile of the uniform distribution is given by $x_p = a + p(b-a)$, where $0 \leq p \leq 1$, representing the value below which a proportion $p$ of the distribution lies
  • Median, the 50th percentile, is calculated as $x_{0.5} = a + 0.5(b-a) = \frac{a+b}{2}$, which is the same as the mean

Exponential Distribution

Exponential distribution and Poisson process

• Exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate
• PDF of the exponential distribution is $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$, and $0$ otherwise, where $\lambda$ is the rate parameter representing the average number of events per unit time
• CDF of the exponential distribution is $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$, giving the probability that an event occurs within a specific time interval
• Mean of the exponential distribution is the reciprocal of the rate parameter, $\mu = \frac{1}{\lambda}$, representing the average time between events
• Variance of the exponential distribution is the square of the reciprocal of the rate parameter, $\sigma^2 = \frac{1}{\lambda^2}$, measuring the spread of the distribution

Applications of exponential distributions

• Exponential distribution exhibits the memoryless property, where the probability of an event occurring in the next time interval is independent of the time that has already passed
  • Mathematically, $P(X > s+t | X > s) = P(X > t)$ for all $s, t \geq 0$, meaning the probability of an event not occurring in the next $t$ time units, given that it has not occurred in the past $s$ time units, is the same as the probability of it not occurring in the first $t$ time units
• Probability calculations using the memoryless property:
  1. $P(X > t) = 1 - F(t) = e^{-\lambda t}$, the probability of an event not occurring within time $t$
  2. $P(X > s+t | X > s) = \frac{P(X > s+t)}{P(X > s)} = \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}} = e^{-\lambda t}$, the probability of an event not occurring in the next $t$ time units, given that it has not occurred in the past $s$ time units
• Applications of the exponential distribution include:
  • Modeling the time between customer arrivals at a service center (bank, store)
  • Analyzing the lifespan of electronic components (light bulbs, batteries)
  • Determining the time between failures in a system (machine breakdowns, server crashes)