Continuous Probability Distributions

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Discrete vs continuous probability distributions, application to normal distributions, use of range from a continuous probability distribution

Last updated 11:33 AM on 6/2/26
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14 Terms

1
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What are the visual differences in plots between discrete and continuous probability distributions?

  • Discrete = jagged

  • Continuous = smooth

<ul><li><p>Discrete = jagged</p></li><li><p>Continuous = smooth</p></li></ul><p></p>
2
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What are the 2 main ways in which CPDs differ from DPDs?

  • P(X = x) = 0

    • The probability of any single value is 0

  • CPDs are described using the probability density function (not probability mass function)

3
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What is the most widely used continuous probability distribution?

Normal distribution AKA Gaussian distribution → is uni-modal (1 peak) + symmetrical

4
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What is the probability density function equation for normal distributions?

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What are the key aspects of a standard normal distribution?

  • Mean = 0

  • SD = 1

<ul><li><p>Mean = 0</p></li><li><p>SD = 1</p></li></ul><p></p>
6
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What are standard normal distributions also known as and why?

z-distributions

  • Presented in terms of z-scores

  • Standardise values of x

    • Numerator: converts x to deviations from the mean

    • Denominator: scales these deviation values based on the SD of the variable

<p>z-distributions</p><ul><li><p>Presented in terms of z-scores</p></li><li><p>Standardise values of x</p><ul><li><p>Numerator: converts x to deviations from the mean</p></li><li><p>Denominator: scales these deviation values based on the SD of the variable</p></li></ul></li></ul><p></p>
7
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What happens when the mean changes in a standard normal distribution?

Adjusts where the curve is centred on the x-axis

<p>Adjusts where the curve is centred on the x-axis</p>
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What happens when the SD changes in a standard normal distribution?

Adjusts the shape of the curve

<p>Adjusts the shape of the curve</p>
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What are the properties of any normal distribution?

  • Around 68% of area falls under 1 SD on either side of mean 

  • Around 95% of area falls under 2 SD on either side of mean 

    • Exactly 95% falls under +/- 1.96 SD 

  • Around 99.75% of area falls under 3 SD on either side of mean 

10
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Integral equation

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11
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How can the integral value (when calculating the area under a curve) be calculated?

Using the PDF

12
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What is the code for calculating the probability density function of the normal distribution?

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What is the code for calculating where x% of the most extreme values in a normal distribution fall?

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How does the t-distribution compare to the normal (z) distribution?

When calculating t, we replace the population SD with the sample SD → tails are slightly higher to account for extra variability from using an estimate (vs actual population value(