Question

The daily exchange rate for currencies fluctuates on a daily basis due to many economic conditions affecting the business cycle. The exchange rate for a twelve month period in the year 2004 between the US dollar and the EURO shows an approximately normally distributed behavior with a mean exchange rate of 0.804 euros for every dollar and a standard deviation of 0.0255.

Find the following:

A) The probability that the exchange rate between the pair of currencies between 0.798 and 0.8100.

B) The probability that the exchange rate will be larger than 0.845 euros for every dollar.

C) The exchange rate such that 98% of the data falls below it.

D) If the standard deviation is changed from the stated value to 0.03, what will the answers in (A) through (C) be.

Answer 1

Let X be the exchange rate for a tweleve month period in 2004.

X~Normal(Mean=0.804, SD=0.0255)

(A) To find P ( 0.798 ≤ X ≤ 0.8100) = P(X≤0.8100) -P(X≥0.798) = P[Z≤ {(0.8100-0.804)/0.0255}] - P(Z≥{(0.798-0.804)/0.0255}] =P(Z≤0.235)-P(Z≥ -0.235) =P(Z≤0.235) - P(Z≤-0.235) (By symmetry property) =0.593-0.407=0.186

(B) P(X>0.845) = 1-P(X≤0.845)=1-P(Z≤{(0.845-0.804)/0.0255)=1-P(Z≤1.608)=1-0.946=0.054

(C) We need to find X for P(X≤x)=0.98

=> P(Z≤(x-0.804)/0.0255)=0.98....(1)

=>P(Z≤z)=0.98

From the normal table we find z =2.054 for 98% of data

So, (1) we have,

(x-0.804)/0.0255=2.054

=>x-0.804=2.054*0.0255

=>x=0.804+2.054*0.0255=0.8564

Hence, the exchange rate is 0.8564.

(D) If SD=0.03 we need to calculate the above,

So (A) To find P ( 0.798 ≤ X ≤ 0.8100) = P(X≤0.8100) -P(X≥0.798) = P[Z≤ {(0.8100-0.804)/0.03}] - P(Z≥{(0.798-0.804)/0.03}] =P(Z≤0.2)-P(Z≥ -0.2) =P(Z≤0.2) - P(Z≤-0.2) (By symmetry property) =0.579-0.421=0.158

(B) P(X>0.845) = 1-P(X≤0.845)=1-P(Z≤{(0.845-0.804)/0.03)=1-P(Z≤1.367)=1-0.914=0.086

C) We need to find X for P(X≤x)=0.98

=> P(Z≤(x-0.804)/0.03)=0.98....(1)

=>P(Z≤z)=0.98

From the normal table we find z =2.054 for 98% of data

So, (1) we have,

(x-0.804)/0.03=2.054

=>x-0.804=2.054*0.03

=>x=0.804+2.054*0.03=0.8656

Hence, the exchange rate is 0.8656.