Probability

Problem Set 1

Part 1:

Please write a function to compute the expected value and standard deviation of an array of values. Compare your results with that of R’s mean and std functions. Please document your work in an R-Markdown file and ensure that you have good comments to help the reader follow your work.

calc_mean = function(calc_vec) {
  
#Expected value
expected_sum =0
expected_mean=0
for(i in 1:length(calc_vec)){
  expected_sum = expected_sum+calc_vec[i]
}

expected_mean = expected_sum/length(calc_vec)
return(expected_mean)

}

calc_var = function(calc_vec){

#Variance
#Expected value
expected_sum =0
expected_mean=0

variance=0
for(i in 1:length(calc_vec)){
  variance = variance +(((calc_vec[i]-expected_mean)^2)/(length(calc_vec)-1))
  
}

return(variance)

}


temperature = c(39,37,41,41,39,42,52,55,35,30,26,29,36,41,39,33,30,27,25,32,43,48,44,35,31,28,26,39,53,58,54,48)

#Calculated mean
calc_mean(temperature)

## [1] 38.625

#Mean from predefined function
mean(temperature)

## [1] 38.625

#Calculated Variance
calc_var(temperature)

## [1] 1626.516

#Variance from pre-defined function

var(temperature)

## [1] 86.5

#Sd from pre-defined function
calc_sd = var(temperature)

sqrt(calc_sd)

## [1] 9.300538

sd(temperature)

## [1] 9.300538

#############

Part 2:

Now, consider that instead of being able to neatly fit the values in memory in an array, you have an infinite stream of numbers coming by. How would you estimate the mean and standard deviation of such a stream? Your function should be able to return the current estimate of the mean and standard deviation at any time it is asked. Your program should maintain these current estimates and return them back at any invocation of these functions. (Hint: You can maintain a rolling estimate of the mean and standard deviation and allow these to slowly change over time as you see more and more new values).

Formulas: Calculating Variance from below two formulas

\[s^2_n = \frac{(n-2)}{(n-1)} \, s^2_{n-1} + \frac{(x_n - \bar x_{n-1})^2}{n}, \quad n>1 \] \[\sigma _{n}^{2}={\frac {(n-1)\,\sigma _{{n-1}}^{2}+(x_{n}-{\bar x}_{{n-1}})(x_{n}-{\bar x}_{{n}})}{n}}\]

#Part 2:

#link https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Online_algorithm

update1 = c(39,37,41,41,39,42,52,55,35,30,26,29,36,41,39,33,30,27,25,32,43,48,44,35,31,28,26,39,53,58,54,48)+20


#Mean and Variance

store =c(0)

new_mean_var = function(old_list,new_list){
  
expected_mean_new = calc_mean(old_list)
variance_new = calc_var(old_list)
count=length(old_list)

for(i in 1:length(new_list)){
  
  count = count+1
  expected_mean_old = expected_mean_new
  
  expected_mean_new= expected_mean_new + ((new_list[i]-expected_mean_new)/count)
  
  #variance_new = variance_new +(((update1[i]-expected_mean_new)^2)/(count-1))
  variance_old = variance_new
  variance_new = (((count-1)*variance_old) +((new_list[i]-expected_mean_old)*(new_list[i] - expected_mean_new)))/count
  
  variance_new1 = (((count-2)*variance_old)/(count-1)) +(((expected_mean_old - expected_mean_new)^2)/count)

}
store[1] = expected_mean_new
store[2] = variance_new
#store[3] = variance_new1

return(store)
}

#Mean
print(new_mean_var(temperature,update1)[1])

## [1] 48.625

#Mean using Standard function
mean(c(temperature,update1))

## [1] 48.625

#Variance
print(new_mean_var(temperature,update1)[2])

## [1] 955.1565

#Variance using Standard function
var(c(temperature,update1))

## [1] 186.7143