Sixth Term Examination Paper (STEP) Probability and Statistics Flashcards

24 sample cards from the Probability and Statistics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Give the pmf, mean and variance of a discrete uniform distribution on $\{1,2,\dots,n\}$.

   $P(X=r)=\dfrac{1}{n}$ for $r=1,\dots,n$. Mean $E(X)=\dfrac{n+1}{2}$; variance $\operatorname{Var}(X)=\dfrac{n^2-1}{12}$.

2. Define the probability generating function $G_X(t)$ of a non-negative integer random variable and how to recover $E(X)$.

   $G_X(t)=E(t^X)=\sum_{r\geq 0}P(X=r)\,t^r$. Then $E(X)=G_X'(1)$ and $\operatorname{Var}(X)=G_X''(1)+G_X'(1)-\big(G_X'(1)\big)^2$, while $P(X=r)=\dfrac{G_X^{(r)}(0)}{r!}$.

3. Define the moment generating function $M_X(t)$ and state how moments are obtained from it.

   $M_X(t)=E(e^{tX})$. The $n$th moment is $E(X^n)=M_X^{(n)}(0)$, the $n$th derivative evaluated at $t=0$. In particular $E(X)=M_X'(0)$ and $E(X^2)=M_X''(0)$.

4. How does the mgf behave for a sum of independent variables and under a linear transformation?

   If $X,Y$ are independent, $M_{X+Y}(t)=M_X(t)\,M_Y(t)$. For constants $a,b$, $M_{aX+b}(t)=e^{bt}M_X(at)$.

5. For a continuous random variable, define the probability density function $f(x)$ and the conditions it satisfies.

   $f(x)\geq 0$ for all $x$, and $\displaystyle\int_{-\infty}^{\infty} f(x)\,dx=1$. Probabilities come from areas: $P(a\leq X\leq b)=\displaystyle\int_a^b f(x)\,dx$, and $P(X=x)=0$ for any single point.

6. State the relationship between the pdf $f(x)$ and the cdf $F(x)$ of a continuous variable.

   $F(x)=\displaystyle\int_{-\infty}^{x} f(u)\,du$ and, where $F$ is differentiable, $f(x)=F'(x)=\dfrac{dF}{dx}$.

7. Give the formulas for $E(X)$ and $\operatorname{Var}(X)$ of a continuous random variable with pdf $f(x)$.

   $E(X)=\displaystyle\int_{-\infty}^{\infty} x f(x)\,dx$ and $\operatorname{Var}(X)=\displaystyle\int_{-\infty}^{\infty}(x-\mu)^2 f(x)\,dx=E(X^2)-\mu^2$, where $E(X^2)=\int x^2 f(x)\,dx$.

8. How are the median and mode of a continuous random variable found from $f$ and $F$?

   The median $m$ satisfies $F(m)=\dfrac{1}{2}$, i.e. $\displaystyle\int_{-\infty}^{m} f(x)\,dx=\tfrac12$. The mode is the value of $x$ maximising the pdf $f(x)$, found where $f'(x)=0$ (a maximum) within the range.

9. Give the pdf, mean and variance of the continuous uniform distribution on $[a,b]$.

   $f(x)=\dfrac{1}{b-a}$ for $a\leq x\leq b$. Mean $E(X)=\dfrac{a+b}{2}$; variance $\operatorname{Var}(X)=\dfrac{(b-a)^2}{12}$.

10. Give the pdf, cdf, mean and variance of the exponential distribution with rate $\lambda$.

    $f(x)=\lambda e^{-\lambda x}$ for $x\geq 0$; $F(x)=1-e^{-\lambda x}$. Mean $E(X)=\dfrac{1}{\lambda}$; variance $\operatorname{Var}(X)=\dfrac{1}{\lambda^2}$. It is memoryless.

11. Give the pdf of the normal distribution $X\sim N(\mu,\sigma^2)$ and its key shape properties.

    $f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)$. It is symmetric and bell-shaped about $x=\mu$, with mean = median = mode = $\mu$ and variance $\sigma^2$; points of inflection are at $x=\mu\pm\sigma$.

12. How do you standardise a normal variable, and what is the standard normal distribution?

    If $X\sim N(\mu,\sigma^2)$ then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)$, the standard normal. Probabilities are read using $\Phi(z)=P(Z\leq z)$, with $\Phi(-z)=1-\Phi(z)$.

13. State the approximate proportions of a normal distribution within $1$, $2$ and $3$ standard deviations of the mean.

    About $68\%$ lies within $\mu\pm\sigma$, about $95\%$ within $\mu\pm 2\sigma$ (more precisely $1.96\sigma$), and about $99.7\%$ within $\mu\pm 3\sigma$ (the empirical/68–95–99.7 rule).

14. Describe the cdf method for finding the distribution of $Y=g(X)$ for a monotonic increasing $g$.

    Compute $F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))=F_X\big(g^{-1}(y)\big)$, then differentiate to get $f_Y(y)=f_X\big(g^{-1}(y)\big)\left|\dfrac{dx}{dy}\right|$.

15. State the change-of-variable (Jacobian) formula for the pdf of $Y=g(X)$ when $g$ is monotonic.

    $f_Y(y)=f_X\big(g^{-1}(y)\big)\left|\dfrac{d}{dy}g^{-1}(y)\right|$. The absolute value of the derivative ensures the resulting density is non-negative.

16. Distinguish the mean, median and mode as measures of location.

    The mean is the arithmetic average $\bar{x}=\dfrac{1}{n}\sum x_i$ (sensitive to outliers). The median is the middle value when ordered (robust to outliers). The mode is the most frequently occurring value. For a symmetric distribution they coincide; for right-skew, mean > median > mode.

17. Define range, interquartile range, variance and standard deviation as measures of spread.

    Range $=$ max $-$ min. Interquartile range $\mathrm{IQR}=Q_3-Q_1$. Variance $s^2=\dfrac{1}{n}\sum(x_i-\bar x)^2$ (or with $n-1$ for a sample estimate). Standard deviation $s=\sqrt{s^2}$, in the same units as the data.

18. How are outliers commonly identified using the IQR, and what does skewness indicate?

    An outlier is typically a value below $Q_1-1.5\,\mathrm{IQR}$ or above $Q_3+1.5\,\mathrm{IQR}$. Positive (right) skew has a long upper tail with mean > median; negative (left) skew has a long lower tail with mean < median.

19. State the distribution of the sample mean $\bar X$ for a sample of size $n$ from a population with mean $\mu$ and variance $\sigma^2$.

    $E(\bar X)=\mu$ and $\operatorname{Var}(\bar X)=\dfrac{\sigma^2}{n}$, so the standard error is $\dfrac{\sigma}{\sqrt{n}}$. If the population is normal, $\bar X\sim N\!\left(\mu,\dfrac{\sigma^2}{n}\right)$ exactly.

20. State the Central Limit Theorem.

    For independent identically distributed variables with mean $\mu$ and finite variance $\sigma^2$, the sample mean $\bar X$ is approximately $N\!\left(\mu,\dfrac{\sigma^2}{n}\right)$ for large $n$, regardless of the population distribution. Equivalently $\dfrac{\bar X-\mu}{\sigma/\sqrt{n}}\to N(0,1)$.

21. What is an unbiased estimator, and give unbiased estimators of the population mean and variance.

    An estimator $\hat\theta$ is unbiased if $E(\hat\theta)=\theta$. The sample mean $\bar X$ is unbiased for $\mu$. The sample variance $s^2=\dfrac{1}{n-1}\sum(X_i-\bar X)^2$ is an unbiased estimator of $\sigma^2$ (the $n-1$ divisor corrects the bias).

22. Give the formula for a $100(1-\alpha)\%$ confidence interval for a mean with known variance $\sigma^2$.

    $\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$, where $z_{\alpha/2}$ is the standard normal critical value (e.g. $1.96$ for $95\%$, $2.576$ for $99\%$). The interval widens with larger confidence and narrows with larger $n$.

23. In hypothesis testing, define the null and alternative hypotheses, the significance level, and Type I and Type II errors.

    $H_0$ is the default claim being tested; $H_1$ is the alternative. The significance level $\alpha$ is $P(\text{reject }H_0\mid H_0\text{ true})$. A Type I error rejects a true $H_0$ (probability $\alpha$); a Type II error fails to reject a false $H_0$ (probability $\beta$); the power is $1-\beta$.

24. Define the product moment correlation coefficient $r$ and the equation of the least-squares regression line of $y$ on $x$.

    $r=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ with $-1\leq r\leq 1$, measuring linear association. The regression line of $y$ on $x$ is $y-\bar y=b(x-\bar x)$ with gradient $b=\dfrac{S_{xy}}{S_{xx}}$, where $S_{xy}=\sum xy-\dfrac{(\sum x)(\sum y)}{n}$ and $S_{xx}=\sum x^2-\dfrac{(\sum x)^2}{n}$.

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