Continuous Probability Distributions: Uniform and Exponential Models

Overview of Continuous Distributions

• The study of continuous distributions follows the previous focus on the normal distribution.

• Summary of the three primary continuous distributions covered in this chapter:

  • Normal Distribution: Symmetrical and bell-shaped. It ranges theoretically from negative infinity ($-\infty$) to positive infinity ($+\infty$). In practice, it is often bounded because the probability tails off and becomes so small that it is effectively zero at extremes.

  • Uniform Distribution: Described as a rectangular distribution.

  • Exponential Distribution: Models the time between events.

• The normal distribution is highly significant for second-semester statistical applications, but the uniform and exponential distributions are essential components of this chapter.

The Uniform Distribution

• The shape of a distribution dictates its behavioral characteristics. The uniform distribution is visually represented as a rectangle.

• Graphical Components:

  • The $x$-axis represents the random variable $x$.

  • The $y$-axis represents the probability density function, $f(x)$.

  • The distribution is characterized by a starting value, $a$, and an ending value, $b$.

• Definition of the Probability Density Function (PDF):

  • The height of the function is constant between the bounds $a$ and $b$.

  • The function is defined piecewise:         $\begin{cases} f(x) = \frac{1}{b-a}, & \text{if } a \le x \le b \\ f(x) = 0, & \text{otherwise} \end{cases}$

• Height and Probability:

  • The point where the function sits on the $y$-axis is its height: $\frac{1}{b-a}$.

  • Note that the height of the function, $f(x)$, does not represent probability. In continuous distributions, probability is defined as the area under the function.

  • For a uniform distribution, calculating probability is simpler than for a normal distribution because it involves finding the area of a rectangle ($\text{base} \times \text{height}$) rather than performing complex integration.

Methods for Calculating Uniform Probabilities

Approach A: The Geometric/Intuitive Method

• This approach relies on identifying the base and height of the rectangular segment.

• Example Situation:

  • Given parameters: $a = 2$ and $b = 6$.

  • Height calculation: $f(x) = \frac{1}{6-2} = \frac{1}{4} = 0.25$.

• Probability Calculation Scenarios:

  • Find $P(3 \le x \le 5)$:

    • The base is the distance between $3$ and $5$, which is $2$ units.

    • The height is $0.25$.

    • $P(3 \le x \le 5) = 2 \times 0.25 = 0.5$.

  • Find $P(x \le 3)$:

    • The distribution starts at $a = 2$, so the base is from $2$ to $3$, which is $1$ unit.

    • The height is $0.25$.

    • $P(x \le 3) = 1 \times 0.25 = 0.25$.

Approach B: The Cumulative Distribution Formula Method

• This approach is based on the integration of the probability density function to find an expression for cumulative probability.

• The result of integrating $\frac{1}{b-a}$ between a lower bound $a$ and a point $t$ provides a closed-form solution for the cumulative probability:

  • Formula: P(X < t) = \frac{t-a}{b-a}

• Revisiting the Example ($a=2, b=6$):

  • Find $P(3 \le X \le 5)$:

    • This can be rewritten as: $P(X \le 5) - P(X \le 3)$.

    • Using the formula for $P(X \le 5)$: $\frac{5 - 2}{6 - 2} = \frac{3}{4} = 0.75$.

    • Using the formula for $P(X \le 3)$: $\frac{3 - 2}{6 - 2} = \frac{1}{4} = 0.25$.

    • Result: $0.75 - 0.25 = 0.5$.

  • Find $P(X \le 3)$:

    • $\frac{3 - 2}{6 - 2} = 0.25$.

Descriptive Statistics for Uniform Distributions

• If a random variable $X$ is uniformly distributed with parameters $a$ and $b$:

• Mean (Expected Value):

  • $\mu = E(x) = \frac{a+b}{2}$

  • In the example where $a=2, b=6$, the mean is $\frac{2+6}{2} = 4$. This represents the exact midpoint of the distribution.

• Standard Deviation:

  • $\sigma = \sqrt{\frac{(b-a)^2}{12}}$

  • In the example where $a=2, b=6$, the standard deviation is approximately $1.1547$. This represents the average amount values deviate from the mean.

The Exponential Distribution

• The exponential distribution is closely linked to the Poisson distribution from previous study.

• Poisson vs. Exponential Relationship:

  • Poisson: Models the number of events (discrete) occurring within a fixed interval or area of opportunity.

  • Exponential: Models the length of time or space between those events (continuous).

• Inverse Relationship:

  • There is an inverse relationship between the frequency of events ($\lambda$) and the time between events ($x$).

  • If the number of events increases (high $\lambda$), the time between events must decrease.

  • If the number of events decreases (low $\lambda$), the time between events increases.

• Graphical Representation:

  • The graph shows exponential decay.

  • It is bounded by zero ($0$) at the lower end because time cannot be negative.

  • It is theoretically bounded by positive infinity ($+\infty$) at the upper end.

• Probability Density Function (PDF):

  • $\begin{cases} f(x) = \lambda e^{-\lambda x}, &amp; \text{if } x \ge 0 \\ f(x) = 0, &amp; \text{otherwise} \end{cases}$

Calculating Exponential Probabilities

• To find the probability, one must find the area under the curve between specific points, typically using the cumulative distribution function obtained via integration.

• Cumulative Probability Formula:

  • The probability that $x$ is less than a specific value $t$ is given by:

  • P(X < t) = 1 - e^{-\lambda t}

  • This formula is derived from integrating $\lambda e^{-\lambda x}$ between the bounds of $0$ and $t$.

Worked Example: Computer Breakdowns

• Scenario: A computer breaks down on average twice per week.

• Identified Parameters:

  • Rate of events: $\lambda = 2\, \text{breakdowns/week}$.

  • Random Variable $x$: Time between breakdowns (exponentially distributed).

  • Units: Weekly.

• Question: Find the probability that the computer will not break within three weeks (i.e., time between breakdowns is at least three weeks).

• Calculation:

  • We seek P(X > 3).

  • Since the distribution has no upper bound, we use the complement: 1 - P(X < 3).

  • Sub in the formula for P(X < 3): $1 - (1 - e^{-(2)(3)})$.

  • The ones cancel out, leaving: $e^{-6}$.

  • Numerical Result: Approximately $0.0025$.

Descriptive Statistics for Exponential Distributions

• If a random variable $X$ is exponentially distributed with parameter $\lambda$:

• Mean (Expected Value):

  • $\mu = E(x) = \frac{1}{\lambda}$

  • This highlights the inverse relationship. For example, if the average rate is $2$ events per week, the average time between events is $0.5$ weeks.

• Variance:

  • $\sigma^2 = \frac{1}{\lambda^2}$

  • Note from Lecturer: Ensure to include the square in the denominator for variance; some slide materials may contain errors omitting the square.

Questions & Discussion

• Student Question: Why was the "less than" sign (<) used instead of "less than or equal to" ($\le$) in the probability calculation?

• Lecturer Response: In continuous distributions (Chapter 6), the distinction between "strictly less than" and "less than or equal to" does not matter. The probability of the variable equaling exactly one point ($P(X=3)$) is effectively zero. Therefore, if a value is strictly less than $3$, it can go up to $2.99999999$, which is effectively the same as $3$.

• Student Question: Does $\lambda$ refer to time?

• Lecturer Response: No, $\lambda$ refers to the average rate of events occurring, exactly as used in the Poisson distribution. It is quoted in terms of average rates of event occurrence, not the length of time itself.

Future Topics and Administrative Notes

• Normal Approximation to the Binomial Distribution: This topic was skipped during this session and will be the focus of the beginning of the next class (Wednesday).

• Formula Sheet: A formula sheet has been made available. Future lecture time will be dedicated to going through it line by line to explain what is included and what is not.

• Course Overview: The final classes will include big-picture overviews and preparations for upcoming assessments.

• Safety Warning: Due to extreme weather conditions (wind), students are advised to avoid walking or parking under trees to prevent injury or damage from falling branches or trees. Students should avoid long-distance driving if possible.