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-: Hello folks.
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In this lesson, we are going to talk
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about the student's T distribution and its characteristics.
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Before we begin, we use the lowercase letter T
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to define a student's T distribution,
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followed by a single parameter in parenthesis,
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called degrees of freedom.
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We read this next statement
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as variable Y follows a student's T distribution
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with three degrees of freedom.
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As we mentioned in the last video,
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it is a small sample size approximation
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of a normal distribution.
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In instances where we would assume a normal distribution,
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were it not for the limited number of observations,
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we use the student's T distribution.
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For instance, the average lap times for the entire season
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of a Formula One race follow a normal distribution.
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But the lap times for the first lap
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of the Monaco Grand Prix would follow
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a student's T distribution.
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Now, the curve of the student's T distribution
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is also bell-shaped and symmetric.
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However, it has fatter tails to accommodate
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the occurrence of values far away from the mean.
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That is because if such a value features in our limited data
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it would be representing a bigger part of the total.
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Another key difference between the student's T distribution
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and the normal one, is that apart from the mean and variance
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we must also define the degrees of freedom
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for the distribution.
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Great job.
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As long as we have at least two degrees of freedom,
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the expected value of a T distribution is the mean, mu.
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Furthermore, the variance of the distribution
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equals the variance of the sample,
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times the number of degrees of freedom,
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over degrees of freedom minus two.
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Overall, the student's T distribution
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is frequently used when conducting statistical analysis.
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It plays a major role when we want to do hypothesis testing
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with limited data,
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since we also have a table summarizing
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the most important values of its CDF.
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Great. Another distribution that is commonly used
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in statistical analysis, is the Chi-squared distribution.
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In the next video, we will explore where we use it,
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and what other distributions it is related to.
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Thanks for watching.
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