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?
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.
To find the LMS (least mean squares) estimate of Θ, we need to calculate the conditional expectation of Θ given the observed value of X.
The conditional expectation of Θ given the observed value of X, denoted as E[Θ|X=x], can be found using the formula:
E[Θ|X=x] = ∫(θ*f_Θ/x(θ/x), dθ)
In this case, we have f_Θ/x(θ/x) = 2*e^(-2*θ). To find the LMS estimate, we need to calculate the conditional expectation using this function.
a) To find the LMS estimate (conditional expectation) of Θ, we calculate:
E[Θ|X=x] = ∫(θ*2*e^(-2*θ), dθ), for θ>=0
To solve this integral, we can use integration by parts. Let's define u = θ and dv = 2*e^(-2*θ)dθ. Then, we have:
du = dθ and v = -e^(-2*θ)
Using the integration by parts formula:
∫(u*dv) = uv - ∫(v*du)
∫(θ*2*e^(-2*θ), dθ) = -θ*e^(-2*θ) - ∫(-e^(-2*θ), dθ)
∫(θ*2*e^(-2*θ), dθ) = -θ*e^(-2*θ) + ∫(e^(-2*θ), dθ)
Using integration by inverse substitution, we can integrate the right-hand side:
∫(e^(-2*θ), dθ) = (-1/2) * e^(-2*θ)
Substituting this back into the previous equation:
∫(θ*2*e^(-2*θ), dθ) = -θ*e^(-2*θ) + (-1/2) * e^(-2*θ) + C
where C is the constant of integration.
Therefore, the conditional expectation of Θ given X=x is:
E[Θ|X=x] = -θ*e^(-2*θ) + (-1/2) * e^(-2*θ) + C
This is the LMS estimate of Θ.