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Applet Exercise Let Y and W have the distributions given
Chapter 4, Problem 87E(choose chapter or problem)
Problem 87E
Applet Exercise Let Y and W have the distributions given in Exercise 4.86.
a Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the distribution of Y .
b Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the χ 2 distribution with 3 degrees of freedom.
c What is the relationship between the .05-quantile of the gamma (α = 1.5, β = 4) distribution and the .05-quantile of the χ 2 distribution with 3 degrees of freedom? Explain this relationship.
Reference
Applet Exercise When we discussed the χ 2 distribution in this section, we presented (with justification to follow in Chapter 6) the fact that if Y is gamma distributed with α = n/2 for some integer n, then 2Y/β has a χ 2 distribution. In particular, it was stated that when α = 1.5 and β = 4, W = Y /2 has a χ 2 distribution with 3 degrees of freedom.
a Use the applet Gamma Probabilities and Quantiles to find P(Y < 3.5).
b Use the applet Gamma Probabilities and Quantiles to find P(W < 1.75). [Hint: Recall that the χ2 distribution with ν degrees of freedom is just a gamma distribution with α = ν/2 and β = 2.]
c Compare your answers to parts (a) and (b).
Questions & Answers
QUESTION:
Problem 87E
Applet Exercise Let Y and W have the distributions given in Exercise 4.86.
a Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the distribution of Y .
b Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the χ 2 distribution with 3 degrees of freedom.
c What is the relationship between the .05-quantile of the gamma (α = 1.5, β = 4) distribution and the .05-quantile of the χ 2 distribution with 3 degrees of freedom? Explain this relationship.
Reference
Applet Exercise When we discussed the χ 2 distribution in this section, we presented (with justification to follow in Chapter 6) the fact that if Y is gamma distributed with α = n/2 for some integer n, then 2Y/β has a χ 2 distribution. In particular, it was stated that when α = 1.5 and β = 4, W = Y /2 has a χ 2 distribution with 3 degrees of freedom.
a Use the applet Gamma Probabilities and Quantiles to find P(Y < 3.5).
b Use the applet Gamma Probabilities and Quantiles to find P(W < 1.75). [Hint: Recall that the χ2 distribution with ν degrees of freedom is just a gamma distribution with α = ν/2 and β = 2.]
c Compare your answers to parts (a) and (b).
ANSWER:
a
Given
The distribution function for the 0.05th quantile of random variable Y is given by,
F(ϕp) = P(Y ≤ ϕp) = 0.05
Calculation
P(Y > ϕp) = 1 - 0.05 = 0.95
The values 𝛼 = 1.5, 𝛽 = 4 and prob = 0.95 are put into Applet to obtain quantile, ϕp = x = 0.70369.