Give the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ,σ, and the corresponding quantile zα of the standard normal distribution.
Question
Give the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ,σ, and the corresponding quantile zα of the standard normal distribution.
Solution
The quantile function, also known as the inverse cumulative distribution function, of a random variable is used to determine the value below which a given percentage of the observations fall. For a normal distribution with mean μ and standard deviation σ, the quantile function is given by:
g(α) = μ + σ*zα
where:
- α is the order of the quantile (the percentage of observations that fall below the quantile),
- μ is the mean of the normal distribution,
- σ is the standard deviation of the normal distribution,
- zα is the α-th quantile of the standard normal distribution (mean 0, standard deviation 1).
So, to find the α-th quantile of a normal distribution, you would:
- Find the α-th quantile of the standard normal distribution, zα.
- Multiply this by the standard deviation of your normal distribution, σ.
- Add the mean of your normal distribution, μ.
This gives you the α-th quantile of your normal distribution.
Similar Questions
Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T .Give the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ,σ, and the corresponding quantile zα of the standard normal distribution. (viii) Find the ML estimate of g(θ). (x) Derive the bias of g(θˆ) as an estimator of g(θ) and use it to provide a bias-corrected estimator g ̃ of g(θ).
Suppose that the c.d.f. of a random variable X is F(x) = 1-e^(-sqrt(x)), x>=0. a) Find the probability density function of X. b) Compute the quantile function of X. c) How can X be generated from a uniform random number generator?
The mean for the standard normal distribution is ____.
For the same model, poduce a normal quantile-quantile plot.
Where does the curve of the normal distribution touch the x-axis?
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