Question

Use R language. question 7

Q6. Hasting's approximation: For 2 > 0, "(z), the cdf of standard normal r.v., can be approximated by the function h(x) = 1 -0.5(1 +212 +2222 +2323 +2424) - 4 with a1 = 0.196854, a2 = 0.115194, az = 0.000344, and a4 = 0.019527. The approxima- tion error may be upper bounded by 2.5 x 10-4. Use a for loop to evaluate h(1.645) and compare the result of pnorm(1.645). Round all your answers to 5 decimal places. Q7. Use Hasting's approximation introduced above to construct a function phi() to compute the cdf of Normal distribution N(Moº) at any specified value x € (-20, +00). The phi() function call must be of the form phi(x,, o), where x is any real-valued scalar. You are NOT allowed to use any loop or use a vector to store the constants. Evaluate phi(-1,0,1) and compare it with pnorm(-1,0,1). Report the absolute differ- ence between the two.

Answer 1

[1] "phi at -1,0,1" [1] 0.1588762 [1] "pnorm at -1,0,1" [1] 0.1586553 [1] "Absolute difference: [1] 0.0002209108 [1] "2.209108-04"

Code with comments

z=(x-mu)/sigma is used to convert it to standard normal distribution

z = Min 1 phi function(x,mu,sigma) { 2 (x-mu)/sigma 3 sign = z<0 #sign is negative 4. z = abs(z) #for using formula, z>=0 5 al=0.196854 #constants 6 a 2=0.115194 7 a 3=0.000344 8 a4=0.019527 9 h = 1-0.5*((1+a1*z+a2*z**2+a3*78*3+a4*z**4)**(-4)) #apply formula 19 if(sign) { 11 h = 1-h #using normal distribution properties, area till -z is same as (1-area till z) 12 } 13 return(h) 14 } 15 print("phi at -1,0,1") #testing function 16 print(phi(-1,0,1)) 17 print("pnorm at -1,0,1") 18 print (pnorm(-1,0,1)) 19 print("Absolute difference: ") 20 diff = abs(pnorm(-1,0,1)-phi(-1,0,1)) 21 print (diff) 22 format(diff, scientific=TRUE)

phi = function(x,mu,sigma) {
z = (x-mu)/sigma
sign = z<0 #sign is negative
z = abs(z) #for using formula, z>=0
a1=0.196854 #constants
a2=0.115194
a3=0.000344
a4=0.019527
h = 1-0.5*((1+a1*z+a2*z**2+a3*z**3+a4*z**4)**(-4)) #apply formula
if(sign){
h = 1-h #using normal distribution properties, area till -z is same as (1-area till z)
}
return(h)
}
print("phi at -1,0,1") #testing function
print(phi(-1,0,1))
print("pnorm at -1,0,1")
print(pnorm(-1,0,1))
print("Absolute difference: ")
diff = abs(pnorm(-1,0,1)-phi(-1,0,1))
print(diff)
format(diff, scientific=TRUE)