Using Uniform distribution,
a) the mean of distribution is 75.25 and variance is 13.02.
b) the cdf of depth is (x - 7.5)/12.5 , 7.5<x<19 or 1 if X>19
A random variable X is said to have a continuous rectangular distribution over an interval (a, b), i.e.
(−∞<a<b<∞)
if its probability density function is given by,
F(x) = 1/(b-a) , a<x<b
Let the random variable X represents the depth of the bioturbation layer in sediment in a certain region. From the given information the random variable X is uniformly distributed on the interval [7.5, 19].
mean of the uniform distribution is,
μ=E(X)=(b+a)/2
Let the random variable X follows uniform distribution with parameters (a, b), then the variance of the uniform distribution is,
σ²=V(X)=(b−a)²/12
we have, a = 7.5 , b = 19
a) mean of the uniform distribution( μ) = 19 +(7.5)²
= 75.25
variance of the uniform distribution = (19-7.5)²/12
= (12.5)²/12 = 13.02
b) The cumulative distribution function of the random variable X is,
Fₓ(x) = P(X<x)
=> Fₓ(x) = ₇.₅∫ˣ f(x) dx
=> Fₓ(x) = ₇.₅∫ˣ(1/12.5)dx
=> Fₓ(x) = 1/12.5 ₇.₅[x]ˣ = 1/12.5(x - 7.5)
=>Fₓ(x) = (x - 7.5)/12.5
Therefore, the cumulative distribution function of the depth is,
Fₓ(x) = (x - 7.5)/12.5 , 7.5<x<19 or 1 if X>19 or zero if X<7.5
Hence, we get the mean of distribution is 75.5 and variance is 13.02.
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