Normal distribution is the most widely used probability distribution in the financial industry. It is a bell-shaped curve and mean, median mode is the same for normal distribution. It is denoted by  where  is the mean and  is the variance of the sample. If the mean is 0 and variance is 1 then the normal distribution is known as standard normal distribution N(1, 0).

Now let us discuss the main...

Select the sample where all the observation in the population has an equal chance. It can be done in two ways, one without replacement and the other with replacement.

A random sample without replacement can be done by executing the following code:

> RandomSample <- Sampledata[sample(1:nrow(Sampledata), 10,  
>+ replace=FALSE),] 

This generates the...

mean is the equal weightage average of the sample. For example, we can compute the mean of Volume in the dataset Sampledata by executing the following code, which gives the arithmetic mean of the volume:

mean(Sampledata$Volume) 

Median is the mid value of the matrix when it is arranged in a sorted way, which can be computed by executing the following code:

median(Sampledata$Volume) 

Mode is the value present in the attribute which has maximum frequency. For mode, there does not exist an inbuilt...

The null hypothesis is given by  where is the hypothesized lower bound of the population mean.

Let us assume a scenario where an investor assumes that the mean of daily returns of a stock since inception is greater than $10. The average of 30 days' daily return sample is $9.9. Assume the population standard deviation is 0.011. Can we reject the null hypothesis at .05 significance level?

Now let us calculate the test statistics z which can be computed by the following code in R:

> xbar= 9.9           
> mu0 = 10            
> sig = 1.1            
> n = 30                  
> z = (xbar-mu0)/(sig/sqrt(n))  
> z  

Here:

Maximum likelihood estimation (MLE) is a method for estimating model parameters on a given dataset.

Now let us try to find the parameter estimates of a probability density function of normal distribution.

Let us first generate a series of random variables, which can be done by executing the following code:

> set.seed(100) 
> NO_values <- 100 
> Y <- rnorm(NO_values, mean = 5, sd = 1) 
> mean(Y) 

This gives 5.002913.

> sd(Y) 

This gives 1.02071.

Now let us make a function for log likelihood:

LogL <- function(mu, sigma) { 
+      A = dnorm(Y, mu, sigma) 
+      -sum(log(A)) 
+  } 

Now let us apply the function mle to estimate the parameters for estimating mean and standard deviation:

  > library(stats4) 
> mle(LogL, start = list(mu = 2, sigma=2)) 

mu and sigma have been...

Let us construct a boxplot for the variable volume of the Sampledata, which can be done by executing the following code:

> boxplot(Sampledata$Volume, main="Volume", boxwex=0.1) 

The graph is as follows:

Figure 2.16: Boxplot for outlier detection

An outlier is an observation which is distant from the rest of the data. When reviewing the preceding boxplot, we can clearly see the outliers which are located outside the fences (whiskers) of the boxplot.

The local outlier factor (LOF) is used for identifying density-based local outliers. In LOF, the local density of a point is compared with that of its neighbors. If the point is in a sparser region than its neighbors then it is treated as an outlier. Let us consider some of the variables from...