To find the LMS estimate (conditional expectation) of Θ given a particular value x of X, we need to find E[Θ|x].
The conditional density function f_Θ/x(θ/x) is given as 2*e^(-2*θ) for θ >= 0.
To find E[Θ|x], we need to integrate θ times this conditional density function over the range of possible values of Θ, which is [0, ∞).
E[Θ|x] = ∫(0 to ∞) (θ * f_Θ/x(θ/x)) dθ
= ∫(0 to ∞) (θ * 2*e^(-2*θ)) dθ
To evaluate this integral, we can use integration by parts. Let u = θ and dv = 2*e^(-2*θ) dθ.
du = dθ and v = -e^(-2*θ) / 2
Using the integration by parts formula:
∫ u dv = uv - ∫ v du
= -θ * (e^(-2*θ) / 2) + ∫ (e^(-2*θ) / 2) dθ
= -θ * (e^(-2*θ) / 2) - (e^(-2*θ) / 4)
Now, we can evaluate the integral:
∫(0 to ∞) (θ * 2*e^(-2*θ)) dθ
= [-θ * (e^(-2*θ) / 2) - (e^(-2*θ) / 4)] evaluated from 0 to ∞
Taking the limit as θ approaches ∞, we have:
lim (θ→∞) [-θ * (e^(-2*θ) / 2) - (e^(-2*θ) / 4)]
= lim (θ→∞) -θ * (e^(-2*θ) / 2) - lim (θ→∞) (e^(-2*θ) / 4)
Both of these limits approach zero, so the expression becomes:
= 0 - 0
Therefore, the LMS estimate (conditional expectation) of Θ given a particular value x of X is 0.
Let Θ and X be jointly continuous nonnegative random variables. A particular value x of X is observed and it turns out that f_Θ/x(θ/x) = 2*e^(-2*θ), for θ>= 0.
a) What's the LMS estimate (conditional espectation) of Θ?
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