Basic Statistical Properties of Data

Abstract

This chapter starts with some basic exploratory statistical properties from sample data. Concept of moment and expectation, and moment-generating and characteristic functions are considered afterwards. Different methods for parameter estimation build the foundation for many statistical inferences in the field of hydrology and hydroclimatology.

Exercise

3.1

Considering the number of storms in an area for the month of June to follow the following distribution

Evaluate the mean and median for the number of storms in the given month. (Ans: 0.848; median lies between 1 and 2)

3.2

Soil samples are collected from 15 vegetated locations in a particular area. The moisture content of the samples as obtained from the laboratory tests is shown in the following table. Evaluate the arithmetic mean, geometric mean, range, variance, coefficient of skewness, and coefficient of kurtosis of the soil moisture data. Comment regarding the skewness and kurtosis of the data. (Ans: 0.3207; 0.2926; 0.490; 0.018; 0.4136; 3.496)

Sample no	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
SMC	0.25	0.40	0.11	0.45	0.36	0.24	0.26	0.31	0.50	0.60	0.39	0.28	0.19	0.14	0.33

3.3

The maximum temperature (in $^{\circ }$C) at a city in the month of May follows the distribution as given below

Evaluate the mean, variance, and coefficient of variation of the maximum temperature in the city. (Ans: 42.5 $^{\circ }$C; 2.083; 0.034)

3.4

The discharge at a gauging station follows the given distribution

Determine the nature of the distribution in terms of its coefficient of variation, skewness, and tailedness. (Ans: 1/5; 2; 6)

3.5

A city supplied water from two sources. The joint pdf of discharge from two sources is as follows:

Evaluate the marginal probability density of each source and the mean discharge from the two sources. Also, evaluate the covariance and the forms of conditional distribution of X given $Y = y$. (Ans: 13/18 units; 30/27 units; −1/162)

3.6

Consider a random variable X to follow a two-parameter distribution. The population mean ($\mu$) and standard deviation ($\sigma$) are the parameters of the distribution. Evaluate an unbiased estimation of $\mu$ and unbiased and biased estimation of $\sigma$.

3.7

Let $x_1,x_2,\dots , x_n$ be a random sample for a distribution with pdf

Find estimators for $\alpha$ and $\beta$ using method of moments.

3.8

Let $x_1,x_2,\dots , x_n\sim U(0,\theta )$. Find the maximum-likelihood estimate of $\theta$?

3.9

If $x_1,x_2,\dots , x_n\sim \frac{e^{-\lambda } \lambda ^{x} }{x!}$. Find the maximum-likelihood estimate of $\lambda$?

3.10

Considering the peak annual discharge at a location to have a mean of 1100 cumec and standard deviation of 260 cumec. Without making any distributional assumptions regarding the data, what is the probability that the peak discharge in any year will deviate more than 800 cumec from the mean? (Ans: 0.106)

3.11

The random variable X can assume the values 1 and −1 with probability 0.5 each. Find (a) the moment-generating function and (b) the first four moments about the origin. (Ans: (a) $E(e^{tX} )=\frac{1}{2} (e^{t} +e^{-t} )$, (b) 0, 1, 0, 1)

3.12

A random variable X has density function given by

Find (a) the moment-generating function and (b) the first four moments about the origin. (Ans: (b) 1/2, 1/2, 3/4, 3/2)

3.13

Find the first four moments (a) about the origin and (b) about the mean, for a random variable X having density function

(Ans: (a) 8/5, 3, 216/35, 27/2 (b) 0, 11/25, 32/875, 3693/8750)

3.14

Find (a) E(X), (b) E(Y), (c) E(X, Y), (d) $E(X^{2})$, (e) $E(Y^{2})$, (f) $\text {Var}(X)$, (g) $\text {Var}(Y)$, (h) $\text {Cov}(X, Y)$ if the joint pdf of random variables X and Y is given as

Use $c=1/210.$ (Ans: (a) 268/63, (b) 170/63, (c) 80/7, (d) 1220/63, (e) 1175/126, (f) 5036/3969, (g) 16225/7938, (h) −200/3969)

3.15

Joint distribution between two random variables X and Y is given as follows:

Find the conditional expectation of (a) Y given X and (b) X given Y. (Ans: (a) $\frac{2x}{3}$ (b) $\frac{2(1+y+y^{2} )}{3(1+y)}$)

3.16

The density function of a continuous random variable X is

Find the (a) mean, (b) median, and (c) mode. (Ans: (a) 1.6 (b) 1.62 (c) 1.73)

3.17

Find the coefficient of (a) skewness and (b) kurtosis of the standard normal distribution which is defined by

(Ans: (a) 0, (b) 3).