Question

Let 𝑋|Λ have an exponential distribution with parameter Λ. Let Λ have a gamma distribution. Determine the unconditional distribution of 𝑋. 𝑋|Λ ~ exp(1/λ) λ ~ Gamma(θ,α)

          Let 𝑋|Λ have an exponential distribution with parameter Λ.
Let Λ have a gamma distribution. Determine the unconditional distribution of 𝑋.
𝑋|Λ ~ exp(1/λ)
λ ~ Gamma(θ,α)
        

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Let 𝑋|Λ have an exponential distribution with parameter Λ. Let Λ have a gamma distribution. Determine the unconditional distribution of 𝑋. 𝑋|Λ ~ exp(1/λ) λ ~ Gamma(θ,α)
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Transcript

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00:01 Let us look at the question that we have been given.
00:02 So we have been given that x lambda has an exponential distribution with parameter lambda, thus the conditional distribution for x given lambda will be f of x lambda is equal to lambda e raised 2 minus lambda x for x greater than 0, 0 otherwise.
00:26 Right.
00:28 Okay.
00:28 Now for the distribution of lambda, that is f of lambda, we will have 1 by 4 lambda greater than 0 and 0 otherwise...
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