If X ∼χ10^2, find the constants a and b so that ℙ(a<X<b)=0.90 and ℙ(X<a)=0.05.

Name: Problem 4, Chapter 6: Probability and Statistics with R
Uploaded: 2020-07-28T19:09:21-08:00
Duration: 4 min 59 s
Channel: Kira Schwander
Description: If X ∼χ10^2, find the constants a and b so that ℙ(a<X<b)=0.90 and ℙ(X<a)=0.05.

If X ∼χ10^2, find the constants a and b so that...

Key Concepts

Chi-square Distribution

The chi-square distribution is a continuous probability distribution that arises from the sum of the squares of independent standard normal random variables. It is widely used in statistics, particularly in hypothesis testing and confidence interval estimation, especially for variance and goodness-of-fit tests.

Degrees of Freedom

Degrees of freedom in the context of the chi-square distribution represent the number of independent pieces of information used to calculate a statistic. In a chi-square distribution, this is usually the number of independent squared standard normal variables summed to create the distribution, and it crucially affects the shape of the distribution.

Quantile Function

The quantile function, also known as the inverse cumulative distribution function, determines the value below which a given percentage of observations in a probability distribution falls. It is used to find critical values corresponding to specific tail probabilities, such as the 5th and 95th percentiles in this problem.

Confidence Interval for a Distribution

A confidence interval in the context of a probability distribution is a range defined by two quantiles that capture a desired proportion of the probability mass. For example, to capture 90% of the data, the lower bound is set to the 5th percentile and the upper bound to the 95th percentile, ensuring that the interval contains the central 90% of the distribution.

Transcript

00:01 We're going to be working on finding a solution in terms of a and b for this inequality.

00:05 So we want to see where these two equations cross each other so that we can see when our endpoints or our boundaries for this inequality are going to be.

00:17 So let's start off by rewriting these as x squared minus 2 a x plus a squared minus x minus b squared, which is x squared plus 2bx, b squared as equal to a squared minus 2 a b plus b squared divided by four so this left side of the equation can be written as a single straight line because we're crossing out these x squareds because they cancel each other out so this line is going to have a slope of negative 2 times a minus b plus a squared minus b squared so this means that because we know that a is greater than b, this number is always going to be a positive number.

01:12 So our slope is always going to be negative for x for this left side.

01:17 So i'm going to write this left side in red.

01:22 And so we're going to have some sort of downward sloping line in our solution for this left side.

01:29 Okay, so keep that in mind.

01:30 And the right side, well, we're going to have some set a and b by the end of this.

01:37 So it's going to be a horizontal line in the y section, so y equals a minus b squared divided by 4.

01:49 And we're looking for the solution that crosses right here at this intersection, so that's going to be x equals solution.

01:59 So that's what we're looking for here...

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