4. Let I(a) = √ xª-¹e-ªdx be the Gamma function. Then, it is easy to show that I'(a + 1) = al(a). Suppose that the distribution of the life time for a product has the form f(x) = cxx-¹e-x/x е " for x > 0 for two given parameters a and A (both strictly positive). The distribution will be called the Gamma distribution with shape parameter a and scale parameter A. (a) Determine the value of c for which f(x) is a legitimate pdf. (Recall that for f(x) to be legitimate pdf, it needs to be non-negative and integrate to 1.) (b) Find the E(X) and var(X).
4. Let I(a) = √ xª-¹e-ªdx be the Gamma function. Then, it is easy to show that I'(a + 1) = al(a). Suppose that the distribution of the life time for a product has the form f(x) = cxx-¹e-x/x е " for x > 0 for two given parameters a and A (both strictly positive). The distribution will be called the Gamma distribution with shape parameter a and scale parameter A. (a) Determine the value of c for which f(x) is a legitimate pdf. (Recall that for f(x) to be legitimate pdf, it needs to be non-negative and integrate to 1.) (b) Find the E(X) and var(X).
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:4. Let I'(a)=√xª-¹e-ªdx be the Gamma function. Then, it is easy to show that I'(a + 1) = aÃ(a).
Suppose that the distribution of the life time for a product has the form
for x ≥ 0
ƒ(x) = cxª-¹e¯x/λ,
е
for two given parameters a and A (both strictly positive). The distribution will be called the Gamma
distribution with shape parameter a and scale parameter \.
(a) Determine the value of c for which f(x) is a legitimate pdf. (Recall that for f(x) to be legitimate
pdf, it needs to be non-negative and integrate to 1.)
(b) Find the E(X) and var(X).
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