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Chapter 3: Problem 17

Compute the measures of skewness and kurtosis of a distribution which is \(N\left(\mu, \sigma^{2}\right) .\) See Exercises \(1.9 .14\) and \(1.9 .15\) for the definitions of skewness and kurtosis, respectively.

Short Answer

Expert verified
The skewness of the given normal distribution is 0 and the kurtosis is 3.

Step by step solution

01

Definition of Skewness for a Normal Distribution

Skewness of a normal distribution is formulated as: \( Skewness = E\left(\dfrac{X - \mu}{\sigma}\right)^3 \). However, by definition, a normal distribution is symmetric about the mean. This means that it has zero skewness.
02

Definition of Kurtosis for a Normal Distribution

Kurtosis of a distribution is the expectation of the standardized random variable to the power of 4: \( Kurtosis = E\left(\dfrac{X - \mu}{\sigma}\right)^4 \). For a normal distribution, this is equal to 3.

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Most popular questions from this chapter

Suppose \(\mathbf{X}\) has a multivariate normal distribution with mean 0 and covariance matrix $$ \boldsymbol{\Sigma}=\left[\begin{array}{llll} 283 & 215 & 277 & 208 \\ 215 & 213 & 217 & 153 \\ 277 & 217 & 336 & 236 \\ 208 & 153 & 236 & 194 \end{array}\right] $$ (a) Find the total variation of \(\mathbf{X}\). (b) Find the principal component vector Y. (c) Show that the first principal component accounts for \(90 \%\) of the total variation. (d) Show that the first principal component \(Y_{1}\) is essentially a rescaled \(\bar{X}\). Determine the variance of \((1 / 2) \bar{X}\) and compare it to that of \(Y_{1}\). Note that the \(\mathrm{R}\) command eigen(amat) obtains the spectral decomposition of the matrix amat.

The approximation discussed in Exercise \(3.2 .8\) can be made precise in the following way. Suppose \(X_{n}\) is binomial with the parameters \(n\) and \(p=\lambda / n\), for a given \(\lambda>0 .\) Let \(Y\) be Poisson with mean \(\lambda\). Show that \(P\left(X_{n}=k\right) \rightarrow P(Y=k)\), as \(n \rightarrow \infty\), for an arbitrary but fixed value of \(k\). Hint: First show that: $$ P\left(X_{n}=k\right)=\frac{\lambda^{k}}{k !}\left[\frac{n(n-1) \cdots(n-k+1)}{n^{k}}\left(1-\frac{\lambda}{n}\right)^{-k}\right]\left(1-\frac{\lambda}{n}\right)^{n} $$

Let \(f(x)\) and \(F(x)\) be the pdf and the cdf, respectively, of a distribution of the continuous type such that \(f^{\prime}(x)\) exists for all \(x\). Let the mean of the truncated distribution that has pdf \(g(y)=f(y) / F(b),-\infty

Let \(X\) have a Poisson distribution with parameter \(m\). If \(m\) is an experimental value of a random variable having a gamma distribution with \(\alpha=2\) and \(\beta=1\), compute \(P(X=0,1,2)\) Hint: Find an expression that represents the joint distribution of \(X\) and \(m\). Then integrate out \(m\) to find the marginal distribution of \(X\).

Let \(X\) be a random variable such that \(E\left(X^{m}\right)=(m+1) ! 2^{m}, m=1,2,3, \ldots\). Determine the mgf and the distribution of \(X\). Hint: Write out the Taylor series \(^{6}\) of the mgf.
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