Sampling Distribution

STA258

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  • STA258 Lecture 04 Raw

• This is the distribution of any Statistic.
• Ex: Standard Error, is the Standard Deviation of the Sampling Distribution.

STA260

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  • STA260 Makeup Lecture

• The distribution of a Statistic.
• Ex: Standard Error
• Illustrating it.
• Rayleigh Distribution.
  • $f_X(x)=\frac{x}{k^2}{e}^{-\frac{x^2}{2k^2}}$ for $x\ge 0$
  • $k$ is scale
  • Mean: $\mu =k\sqrt{\frac{\pi }{2}}$
  • Variance: ${\sigma }^2=\frac{4-\pi }{2}{k}^2$
  • $f_X(x;k)=\frac{2}{25}x{e}^{-\frac{x^2}{25}}$ for $x>0$
    • $\frac{1}{k^2}=\frac{2}{25}$
    • $2k^2=25$
    • $=\sqrt{\frac{25}{2}}=\frac{5}{\sqrt{2}}$
    • Mean:
      • $\mu =k\sqrt{\frac{\pi }{2}}=\frac{5}{\sqrt{2}}\cdot \sqrt{\frac{\pi }{2}}=4.43$
    • Variance:
      • ${\sigma }^2=\frac{(4-\pi )}{2}{\left(\frac{5}{\sqrt{2}}\right)}^2=5.365$
    • By CLT:
      • $\overline{X}\sim N({\mu }_{\overline{X}}=\mu ,{\sigma }_{\overline{X}}^2=\frac{{\sigma }^2}{n})$
        • Sampling Distribution of $\overline{X}$
      • $\overline{X}\sim N({\mu }_{\overline{X}}=4.43,{\sigma }_{\overline{X}}^2=0.5365)$
• Marginal Distribution of Sample Variance
  • Distribution of $S^2$
  • $\frac{(n-1)S^2}{{\sigma }^2}\sim {\chi }_{(n-1)}^2$
  • $S^2\sim \frac{{\sigma }^2}{n-1}{\chi }_{(n-1)}^2$
    • Scaled Chi-Squared
  • $S^2\sim \frac{5.365}{10-1}{\chi }_{(9)}^2$
  • $S^2\sim 0.5961{\chi }_{(9)}^2$
    • Scaling factor of ${\chi }_{(n)}^2$
    • Let $X\sim {\chi }_{(n)}^2$
    • Find the distribution of $kx$ for $k>0$
    • Transformations - Univariate - STA256
    • Let $Y=kx$
    • Inverse Map
      • $x=\frac{y}{k}$
    • Jacobian
      • $\left|J\right|=\left|\frac{\text{differential}x}{\text{differential}y}\right|$
      • $\left|J\right|=\left|\frac{\text{differential}}{\text{differential}y}\frac{y}{k}\right|=\left|\frac{1}{k}\right|=\frac{1}{k}$
    • If $X\sim {\chi }_{(n)}^2$
    • $f_X(x)=\frac{1}{2^{n/2}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{x}^{n-1}{e}^{-\frac{x}{2}}$
    • Find the Probability Density Function of $y$ using the Jacobian technique
    • $f_Y(y)=\left|J\right|f_X(g^{-1}(y))$
      • $=\frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\left(\frac{y}{k}\right)}^{n-1}{e}^{-\frac{\frac{y}{k}}{2}}\left|\frac{1}{k}\right|$
      • $=\underset{\text{Scaling}}{\underbrace{\frac{1}{k}}}\underset{{\chi }_{(n)}^2}{\underbrace{\frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\left(\frac{y}{k}\right)}^{n-1}{e}^{-\frac{y}{2k}}}}$
      • $=\frac{1}{k}{\chi }_{(n)}^2$
    • In general, for $X\sim f(x)$ doesn't always mean $kX\nsim \frac{1}{k}f(x)$
    • Special Cases:
      • $X\sim {\chi }_{(n)}^2⟹kx\sim \frac{1}{k}{\chi }_{(n)}^{2}k>0$
      • $X\sim \mathrm{Exp}(\lambda )⟹kx\sim \mathrm{Exp}(k\lambda )k>0$
        • #tk exercise
      • $X\sim N(\mu ,{\sigma }^2)⟹kx\sim N(k\mu ,k^2{\sigma }^2)$