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Chapter 3: Problem 2

If \(X\) is \(\chi^{2}(5)\), determine the constants \(c\) and \(d\) so that \(P(c

Short Answer

Expert verified
The constants \(c\) and \(d\) correspond to the 2.5th and 97.5th percentiles of the Chi-Square distribution with 5 degrees of freedom, respectively.

Step by step solution

01

Interpret Probability Statement 1

The first statement \(P(X < c) = 0.025\) means the probability that \(X\) is less than \(c\) is 0.025. In terms of Chi-Square distribution, this is equivalent to the 2.5th percentile of the Chi-Square distribution with 5 degrees of freedom.
02

Find Constant c

To find \(c\), look up the value in Chi-Square distribution table or use the Chi-Square inverse cumulative distribution function corresponding to a probability of 0.025 and 5 degrees of freedom.
03

Interpret Probability Statement 2

The second statement \(P(c < X < d) = 0.95\) means the probability that \(X\) is between \(c\) and \(d\) is 0.95. Given that the cumulative distribution function of \(c\) is 0.025, this implies that the cumulative distribution function at \(d\) is 0.975 (since \(0.025 + 0.95 = 0.975\)).
04

Find Constant d

To find \(d\), look up the value in Chi-Square distribution table or use the Chi-Square inverse cumulative distribution function corresponding to a probability of 0.975 and 5 degrees of freedom.

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Most popular questions from this chapter

If $$ \Phi(x)=\int_{-\infty}^{z} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w $$ show that \(\Phi(-z)=1-\Phi(z)\)

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