GATE Statistics Standard discrete and continuous univariate distributions Flashcards

24 sample cards from the Standard discrete and continuous univariate distributions deck

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

1. How is the Exponential distribution related to the Gamma distribution?

   $\text{Exp}(\lambda) = \text{Gamma}(\alpha=1, \beta=\lambda)$. The sum of $n$ i.i.d. $\text{Exp}(\lambda)$ variables is $\text{Gamma}(n, \lambda)$ (an Erlang distribution).

2. Define the Gamma function $\Gamma(\alpha)$ and give its value at integers and at $\tfrac12$.

   $$\Gamma(\alpha) = \int_0^{\infty} x^{\alpha-1}e^{-x}\,dx.$$ For a positive integer $n$, $\Gamma(n) = (n-1)!$; also $\Gamma\!\left(\tfrac12\right) = \sqrt{\pi}$ and $\Gamma(\alpha+1) = \alpha\Gamma(\alpha)$.

3. State the pdf of a Beta random variable $X \sim \text{Beta}(\alpha,\beta)$.

   $$f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0 < x < 1,$$ where $B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$.

4. What are the mean and variance of $X \sim \text{Beta}(\alpha,\beta)$?

   $E(X) = \frac{\alpha}{\alpha+\beta}$ and $\operatorname{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$.

5. State the pdf of a Chi-square random variable with $n$ degrees of freedom, $X \sim \chi^{2}_{n}$.

   $$f(x) = \frac{1}{2^{n/2}\Gamma(n/2)} x^{n/2 - 1} e^{-x/2}, \quad x > 0.$$ It is $\text{Gamma}(\alpha = n/2, \beta = 1/2)$.

6. What are the mean and variance of $X \sim \chi^{2}_{n}$?

   $E(X) = n$ and $\operatorname{Var}(X) = 2n$.

7. How does the Chi-square distribution arise from standard normal variables?

   If $Z_1,\dots,Z_n$ are i.i.d. $N(0,1)$, then $\sum_{i=1}^{n} Z_i^{2} \sim \chi^{2}_{n}$.

8. Define a Student's $t$ random variable in terms of standard normal and chi-square variables.

   If $Z \sim N(0,1)$ and $V \sim \chi^{2}_{n}$ are independent, then $$T = \frac{Z}{\sqrt{V/n}} \sim t_{n}.$$

9. What are the mean and variance of a Student's $t$ distribution with $n$ degrees of freedom?

   $E(T) = 0$ for $n > 1$ (undefined for $n=1$), and $\operatorname{Var}(T) = \frac{n}{n-2}$ for $n > 2$.

10. Define an $F$ random variable in terms of two independent chi-square variables.

    If $U \sim \chi^{2}_{m}$ and $V \sim \chi^{2}_{n}$ are independent, then $$F = \frac{U/m}{V/n} \sim F_{m,n}.$$

11. State the relationship between the $t$ and $F$ distributions.

    If $T \sim t_{n}$, then $T^{2} \sim F_{1,n}$.

12. What is the mean of an $F_{m,n}$ random variable?

    $E(F) = \frac{n}{n-2}$ for $n > 2$ (it does not depend on the numerator degrees of freedom $m$).

13. State the pdf of a standard Cauchy random variable and a key fact about its moments.

    $$f(x) = \frac{1}{\pi(1+x^{2})}, \quad -\infty < x < \infty.$$ The Cauchy distribution has no finite mean or variance; its mgf does not exist. It equals the ratio of two independent $N(0,1)$ variables and $t_1$.

14. State the pdf of a Lognormal random variable: $X = e^{Y}$ where $Y \sim N(\mu, \sigma^{2})$.

    $$f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(\ln x - \mu)^{2}}{2\sigma^{2}}\right), \quad x > 0.$$

15. What are the mean and variance of a Lognormal random variable with parameters $\mu, \sigma^{2}$?

    $E(X) = e^{\mu + \sigma^{2}/2}$ and $\operatorname{Var}(X) = \left(e^{\sigma^{2}} - 1\right)e^{2\mu + \sigma^{2}}$.

16. State the pdf and cdf of a Weibull random variable with shape $k$ and scale $\lambda$.

    $$f(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^{k}}, \quad x \ge 0,$$ and $F(x) = 1 - e^{-(x/\lambda)^{k}}$. When $k=1$ it reduces to the Exponential.

17. State the pdf of a Laplace (double exponential) random variable with location $\mu$ and scale $b$.

    $$f(x) = \frac{1}{2b}\exp\!\left(-\frac{|x-\mu|}{b}\right), \quad -\infty < x < \infty,$$ with $E(X)=\mu$ and $\operatorname{Var}(X)=2b^{2}$.

18. What is the moment generating function of $X \sim \text{Exp}(\lambda)$ (rate $\lambda$), and for which $t$ does it exist?

    $$M_X(t) = \frac{\lambda}{\lambda - t}, \quad t < \lambda.$$

19. For $X \sim \text{Gamma}(\alpha,\beta)$ with rate $\beta$, state the moment generating function.

    $$M_X(t) = \left(\frac{\beta}{\beta - t}\right)^{\alpha}, \quad t < \beta.$$

20. Distinguish between the discrete and continuous classes of distributions in terms of their probability functions.

    Discrete distributions assign positive probability to individual points via a pmf $P(X=x)$ summing to 1; continuous distributions have a density $f(x)$ with $P(X=x)=0$ for any point and $\int f(x)\,dx = 1$. Probabilities for continuous variables come from integrating the pdf over an interval.

21. What is the maximum likelihood estimator (MLE) of $p$ for an i.i.d. sample $X_1,\dots,X_n \sim \text{Bernoulli}(p)$?

    $\hat{p} = \bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i$, the sample proportion of successes.

22. What is the MLE of $\lambda$ for an i.i.d. sample from $\text{Poisson}(\lambda)$?

    $\hat{\lambda} = \bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i$, the sample mean.

23. What is the MLE of $\lambda$ (rate) for an i.i.d. sample from $\text{Exp}(\lambda)$?

    $\hat{\lambda} = \frac{1}{\bar{X}} = \frac{n}{\sum_{i=1}^{n} X_i}$.

24. Compare the relationship between the mean and variance for the Binomial, Poisson, and Geometric distributions.

    Binomial: $\operatorname{Var} = np(1-p) < np = E$ (underdispersed relative to Poisson). Poisson: $\operatorname{Var} = E = \lambda$ (equidispersed). Geometric (on $\{1,2,\dots\}$): $\operatorname{Var} = \frac{1-p}{p^{2}} > \frac{1}{p} = E$ (overdispersed).

What this deck covers

Standard discrete and continuous univariate distributions flashcards FAQ