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== 6.1#h(0.6em)Introductions


    A continuous random variable (usually denoted as X) is a variable that has an infinite number of random values in an interval of numbers. There are many different types of continuous distributions. To be a valid continuous distribution the total area under the curve has to be equal to one and the function’s y-values need to be positive.


    For example, we may have a random variable that is uniformly distributed so we could use the #link("https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Mostly_Harmless_Statistics_(Webb)/06%3A_Continuous_Probability_Distributions/6.02%3A_Uniform_Distribution")[Uniform distribution]that looks like a rectangle. See Figure 6-1.


    #figure(figph[Uniform distribution graph: a horizontal blue segment at height 1/(b-a) spans between a and b, with dashed vertical lines dropping to open circles at a and b on the x-axis, outlining a rectangle.], alt: "Uniform distribution graph: a horizontal blue segment at height 1/(b-a) spans between a and b, with dashed vertical lines dropping to open circles at a and b on the x-axis, outlining a rectangle.", caption: none)


    Figure 6-1


    We may want to model the time it takes customer service to complete a call with the #link("https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Mostly_Harmless_Statistics_(Webb)/06%3A_Continuous_Probability_Distributions/6.03%3A_Exponential_Distribution")[exponential distribution]. See Figure 6-2.


    #figure(figph[Three exponential distribution curves for mu = 4 in blue, mu = 1.2 in green, and mu = 0.5 in magenta over x from 0 to 10; the mu = 0.5 curve starts highest near 2 on the y-axis and drops fastest, while the mu = 4 curve starts lowest and decays slowest toward zero.], alt: "Three exponential distribution curves for mu = 4 in blue, mu = 1.2 in green, and mu = 0.5 in magenta over x from 0 to 10; the mu = 0.5 curve starts highest near 2 on the y-axis and drops fastest, while the mu = 4 curve starts lowest and decays slowest toward zero.", caption: none)


    Figure 6-2


    We may have standardized test scores that follow a bell-shaped curve like the #link("https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Mostly_Harmless_Statistics_(Webb)/06%3A_Continuous_Probability_Distributions/6.04%3A_Normal_Distribution")[Gaussian (Normal) Distribution]. See Figure 6-3.


    #figure(figph[Four normal distribution curves on a grid from x = -4 to 4: mu = 0, sigma = 1 in blue; mu = -2, sigma = 0.5 in green; mu = 0, sigma = 0.2 in red, tall and narrow; and mu = 0, sigma = 3 in cyan, low and wide.], alt: "Four normal distribution curves on a grid from x = -4 to 4: mu = 0, sigma = 1 in blue; mu = -2, sigma = 0.5 in green; mu = 0, sigma = 0.2 in red, tall and narrow; and mu = 0, sigma = 3 in cyan, low and wide.", caption: none)


    Figure 6-3


    #figure(figph[Four overlapping bell-shaped curves centered at zero labeled d.f. = 50 in green, d.f. = 15 in blue, d.f. = 5 in magenta, and d.f. = 3 in red; smaller degrees of freedom give a slightly lower peak and heavier tails.], alt: "Four overlapping bell-shaped curves centered at zero labeled d.f. = 50 in green, d.f. = 15 in blue, d.f. = 5 in magenta, and d.f. = 3 in red; smaller degrees of freedom give a slightly lower peak and heavier tails.", caption: none)


    Figure 6-4


    We may want to model the average time it takes for a component to be manufactured and use the bell-shaped Student t-distribution. See Figure 6-4.


    This is just an introductory course so we are only going to cover a few distributions. If you want to explore more distributions, check out the chart by Larry Leemis at: #link("http://www.math.wm.edu/~leemis/chart/UDR/UDR.html")[http://www.math.wm.edu/~leemis/chart/UDR/UDR.html].


    
=== Very Important


    The probability of an interval between two X values is equal to the area under the density curve between those two #math.equation(block: false, alt: "X")[$X$] values. For a discrete random variable, we can assign probabilities to each outcome. We cannot do this for a continuous random variable. The probability for a single #math.equation(block: false, alt: "X")[$X$] value for a continuous random variable is 0. Thus “” are equivalent to “≤” and “≥.” In other words,


    #math.equation(block: true, alt: "P open parenthesis a ≤ X ≤ b close parenthesis equals P open parenthesis a less than X less than b close parenthesis equals P open parenthesis a ≤ X less than b close parenthesis equals P open parenthesis a less than X ≤ b close parenthesis")[$P ( a ≤ X ≤ b ) = P ( a < X < b ) = P ( a ≤ X < b ) = P ( a < X ≤ b )$]


    since there is no area of a line.


    We now will look at some specific models that have been found useful in practice. Consider an experiment that consists of observing events in a certain time frame, such as buses arriving at a bus stop or telephone calls coming into a switchboard during a specified period. It may then be of interest to place a probability distribution on the actual time of occurrence. In this section, we will tell you which distribution to use in the question.