For the uniformly distributed random variable X with a = 0 and b = θ, the value of the ratio of raw moments (μ'3/μ'4) is.
5 / 4θ
To find the ratio of the raw moments (μ3/μ4) for a uniformly distributed random variable X with a = 0 and b = θ, we start by calculating the raw moments:
Raw moment μn for uniform distribution is given by:
μn = ∫abxnf(x)dx = ∫0θxn(θ-1)dx
Solving this integral gives:
μn = [θ-1xn+1/(n+1)] from x=0 to x=θ
μn = [θ-1(θn+1)/(n+1)] = θn/(n+1)
Using this formula, calculate μ3 and μ4:
μ3 = θ3/4
μ4 = θ4/5
Now, calculate the ratio (μ3/μ4):
(μ3/μ4) = (θ3/4) / (θ4/5)
= (θ3/4) * (5/θ4)
= 5/(θ * 4)
= 5/4θ
For a simplification, assume θ cancels out:
The ratio of raw moments is 5 / 4θ.
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