GATE Statistics Standard discrete and continuous univariate distributions Syllabus

Standard discrete and continuous univariate distributions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Standard discrete and continuous univariate distributions in GATE Statistics, not a summary of it.

1. 1

   Bernoulli Distribution

   1 topic

   • Introduction
2. 2

   Binomial Distribution

   1 topic

   • Introduction
3. 3

   Geometric Distribution

   1 topic

   • Introduction
4. 4

   Negative Binomial Distribution

   1 topic

   • Introduction
5. 5

   Hypergeometric Distribution

   1 topic

   • Introduction
6. 6

   Discrete Uniform Distribution

   1 topic

   • Introduction
7. 7

   Poisson Distribution

   1 topic

   • Introduction
8. 8

   Continuous Uniform Distribution

   1 topic

   • Introduction
9. 9

   Exponential Distribution

   1 topic

   • Introduction
10. 10

    Gamma Distribution

    1 topic

    • Introduction
11. 11

    Beta Distribution

    1 topic

    • Introduction
12. 12

    Weibull Distribution

    1 topic

    • Introduction
13. 13

    Normal Distribution

    1 topic

    • Introduction

Standard discrete and continuous univariate distributions flashcards for GATE Statistics

21 of 50 cards from the Standard discrete and continuous univariate distributions deck — real questions with worked answers.

1. State the probability mass function (pmf) of a Bernoulli random variable $X \sim \text{Bernoulli}(p)$.

   $$P(X=x) = p^{x}(1-p)^{1-x}, \quad x \in \{0,1\},\ 0 \le p \le 1.$$

2. What are the mean and variance of $X \sim \text{Bernoulli}(p)$?

   $E(X) = p$ and $\operatorname{Var}(X) = p(1-p)$.

3. Give the pmf of a Binomial random variable $X \sim \text{Bin}(n,p)$.

   $$P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k}, \quad k = 0,1,\dots,n.$$

4. What are the mean, variance, and moment generating function of $X \sim \text{Bin}(n,p)$?

   $E(X) = np$, $\operatorname{Var}(X) = np(1-p)$, and $M_X(t) = \left(1 - p + p e^{t}\right)^{n}$.

5. If $X_1 \sim \text{Bin}(n_1,p)$ and $X_2 \sim \text{Bin}(n_2,p)$ are independent, what is the distribution of $X_1 + X_2$?

   $X_1 + X_2 \sim \text{Bin}(n_1 + n_2,\, p)$. The Binomial is closed under addition only when the success probability $p$ is the same.

6. State the pmf of a Poisson random variable $X \sim \text{Poisson}(\lambda)$.

   $$P(X=k) = \frac{e^{-\lambda}\lambda^{k}}{k!}, \quad k = 0,1,2,\dots,\ \lambda > 0.$$

7. What are the mean, variance, and mgf of $X \sim \text{Poisson}(\lambda)$?

   $E(X) = \operatorname{Var}(X) = \lambda$, and $M_X(t) = \exp\!\left(\lambda(e^{t} - 1)\right)$.

8. State the Poisson approximation to the Binomial distribution.

   If $n \to \infty$ and $p \to 0$ with $np \to \lambda$ fixed, then $\text{Bin}(n,p) \to \text{Poisson}(\lambda)$. Useful when $n$ is large and $p$ is small.

9. If $X \sim \text{Poisson}(\lambda_1)$ and $Y \sim \text{Poisson}(\lambda_2)$ are independent, what is the distribution of $X+Y$?

   $X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2)$ (the Poisson is closed under addition).

10. State the pmf, mean, and variance of a Geometric random variable $X$ counting the number of trials until the first success (support $\{1,2,\dots\}$).

    $$P(X=k) = (1-p)^{k-1}p,\quad k=1,2,\dots$$ with $E(X) = \frac{1}{p}$ and $\operatorname{Var}(X) = \frac{1-p}{p^{2}}$.

11. State and explain the memorylessness property of the Geometric distribution.

    $P(X > m+n \mid X > m) = P(X > n)$. Having already waited $m$ trials without success does not change the distribution of additional trials needed. It is the only discrete distribution with this property.

12. State the pmf of a Negative Binomial random variable $X$ counting the number of failures before the $r$-th success.

    $$P(X=k) = \binom{k+r-1}{k} p^{r}(1-p)^{k}, \quad k = 0,1,2,\dots$$

13. What are the mean and variance of the Negative Binomial distribution (number of failures before the $r$-th success)?

    $E(X) = \frac{r(1-p)}{p}$ and $\operatorname{Var}(X) = \frac{r(1-p)}{p^{2}}$.

14. State the pmf of a Hypergeometric random variable for drawing $n$ items without replacement from $N$ items containing $K$ successes.

    $$P(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}, \quad \max(0,n-N+K) \le k \le \min(n,K).$$

15. What are the mean and variance of the Hypergeometric distribution with parameters $N$, $K$, $n$?

    $E(X) = n\frac{K}{N}$ and $\operatorname{Var}(X) = n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$. The factor $\frac{N-n}{N-1}$ is the finite population correction.

16. State the pmf, mean, and variance of a Discrete Uniform random variable on $\{1,2,\dots,n\}$.

    $$P(X=k) = \frac{1}{n},\quad k=1,\dots,n,$$ with $E(X) = \frac{n+1}{2}$ and $\operatorname{Var}(X) = \frac{n^{2}-1}{12}$.

17. State the pdf, mean, and variance of a Continuous Uniform random variable $X \sim U(a,b)$.

    $$f(x) = \frac{1}{b-a},\quad a \le x \le b,$$ with $E(X) = \frac{a+b}{2}$ and $\operatorname{Var}(X) = \frac{(b-a)^{2}}{12}$.

18. State the pdf of a Normal random variable $X \sim N(\mu, \sigma^{2})$.

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

19. What is the moment generating function of $X \sim N(\mu,\sigma^{2})$?

    $$M_X(t) = \exp\!\left(\mu t + \frac{\sigma^{2}t^{2}}{2}\right).$$

20. How is a standard normal variable $Z$ obtained from $X \sim N(\mu,\sigma^{2})$, and what is its distribution?

    $Z = \frac{X - \mu}{\sigma} \sim N(0,1)$, with $E(Z)=0$ and $\operatorname{Var}(Z)=1$.

21. If $X_i \sim N(\mu_i, \sigma_i^{2})$ are independent, what is the distribution of $\sum_i a_i X_i$?

    $$\sum_i a_i X_i \sim N\!\left(\sum_i a_i \mu_i,\ \sum_i a_i^{2}\sigma_i^{2}\right).$$ Any linear combination of independent normals is normal.

See more Standard discrete and continuous univariate distributions flashcards →

Planning Standard discrete and continuous univariate distributions for GATE Statistics

Standard discrete and continuous univariate distributions (GATE Statistics) FAQ