Cheat Sheet Module 2

Cheat Sheet Module 2#

*** - denotes IMPORTANT

Estimators#

General Notations

  • \(\hat{}\) (hat) - denotes an estimated value.

  • \(\bar{}\) (bar) - denotes sample mean.

  • \(n\) - denotes sample size

  • \(\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i\)

  • \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\)

Normal Distribution \(x \sim \mathcal{N}(\mu,\sigma^2)\) ***#

  • *** \(\bar{x} \sim \mathcal{N}(\mu,\frac{\sigma^2}{n}) \implies \hat{\mu} = \bar{x}\) (Unbiased)

    • \(\frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1)\)

    • \(\frac{\bar{x} - \mu}{s/\sqrt{n}} \sim t(n-1)\) (read as “t-distribution or Student’s t-distribution with n-1 degrees of freedom)

  • \(s^2 \sim \frac{\sigma^2}{n-1}\chi^2(n-1)\) (“Scaled Chi-squared” Distribution) \(\implies \hat{\sigma^2} = s^2\)

    • \(\hat{\sigma} = \frac{s}{c_4}\) (from chart)

    • \(\hat{\sigma} = \frac{R}{d_2}\) (from chart)

    • *** Biased \(\hat{\sigma} = s\)

Bernoulli Distribution \(x \sim Bernoulli(p)\)#

  • \(\bar{x} \sim \mathcal{N}(p,\frac{p(1-p)}{n}) \implies \hat{p} = \bar{x}\) (Unbiased, for a large sample size \(n\))

  • *** Bernoulli-Binomial relationship

    • If \(x \sim Bernoulli(p)\) and we do \(K\) trials and let \(y = \sum_{i=1}^{K}x_i\) (i.e., the sum will give the number of successes in \(K\) trials)

      • Then, \(y \sim \mathcal{B}(K,p)\)

Binomial Distribution \(x \sim \mathcal{B}(N,p)\)#

  • \(\bar{x} \sim \mathcal{N}(Np,\frac{Np(1-p)}{n}) \implies \hat{p} = \frac{\bar{x}}{N}\) (Unbiased) (for a large sample size \(n\)) (Note: \(N\) is different from \(n\))

Poisson Distribution \(x \sim Poisson(\lambda)\)#

  • \(\bar{x} \sim \mathcal{N}(\lambda,\frac{\lambda}{n}) \implies \hat{\lambda} = \bar{x}\) (Unbiased, for a large sample size \(n\))

p-value of Hypothesis Testing ***#

Assuming Normal Distribution of \(x\)#

  • Testing on \(\mu\)

    • Known \(\sigma^2\) - “One Sample z-test” \(\implies z_{test} = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}\)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu \neq \mu_0\) \(\implies\) Two-sided \(\implies\) p-value = \(2p(Z \geq |z_{test}|)\)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu > \mu_0\) \(\implies\) One-sided (Upper Tail Test) \(\implies\) p-value = \(p(Z \geq z_{test})\)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu < \mu_0\) \(\implies\) One-sided (Lower Tail Test) \(\implies\) p-value = \(p(Z \leq z_{test})\)

    • Unknown \(\sigma^2\) - “One Sample t-test” \(\implies t_{test} = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}\) (remember to use (n-1) degrees of freedom)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu \neq \mu_0\) \(\implies\) Two-sided \(\implies\) p-value = \(2p(T \geq |t_{test}|)\)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu > \mu_0\) \(\implies\) One-sided (Upper Tail Test) \(\implies\) p-value = \(p(T \geq t_{test})\)

      • \(H_0\): \(\mu = \mu_0\); \(H_a\): \(\mu < \mu_0\) \(\implies\) One-sided (Lower Tail Test) \(\implies\) p-value = \(p(T \leq t_{test})\)

  • Testing on \(\sigma^2\)

    • “Chi-squared Test” \(\implies \chi_{test}^2 = \frac{(n-1)s^2}{\sigma_0^2}\)

    • Two-sided \(\implies\) p-value \(= \min(p(\chi^2 > \chi_{test}^2), p(\chi^2 < \chi_{test}^2)) \)

Errors in Hypothesis Testing#

  • p-value \(\leq \alpha\) \(\implies\) Reject \(H_0\)

  • Type-I Error:(Definition) Reject \(H_0\) when it is True

    • \(\alpha\) = \(p\)(Type-I Error)

    • Typically, the \(\alpha\) values are provided as 5% or 10%

Confidence Intervals#

Useful z-score values#

  • Two-sided

    • \(90\% \implies z_{0.05} = \pm 1.645\)

    • \(95\% \implies z_{0.025} = \pm 1.96\)

    • \(99\% \implies z_{0.005} = \pm 2.575\)

  • One-sided Right Tail

    • \(90\% \implies z_{0.1}^{(r)} = 1.28\)

    • \(95\% \implies z_{0.05}^{(r)} = 1.645\)

    • \(99\% \implies z_{0.01}^{(r)} = 2.33\)

  • One-sided Left Tail

    • \(90\% \implies z_{0.1}^{(l)} = -1.28\)

    • \(95\% \implies z_{0.05}^{(l)} = -1.645\)

    • \(99\% \implies z_{0.01}^{(l)} = -2.33\)

Confidence Interval on Variance#

For given error rate (\(\alpha\))

\(\frac{(n-1)s^2}{\chi_{\frac{\alpha}{2}}^2(n-1)} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi_{1-\frac{\alpha}{2}}^2(n-1)}\)

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