Question

question 01) Let X_i , i = 1 , 2 , 3 , … , 10 be normally distributed independent random variables X _i ∼ N ( μ , σ² ). Determine the mean μ = ......................

and the standard deviation σ = ..................... (round to the third decimal place), if it is known that 1/10 ∑¹⁰_i=1 X_i ∼ N ( 1.5 , 0.3 ).

need  μ and σ

Answer 1

X i ∼ N ( μ , σ^2 ).

That is X i ∼ N ( 1.5 , 0.3 ).

Mean = 1.5

Standard deviation = answer.

Example to calculate mean and standard deviation from 10 random variables:

Sample : {1,2,3,4,5,6,7,8,9,10}

Mean = sum of Xi / total variables

Mean = 5.5

(formula used for sum : sum of 1st n natural number that 1+2+3.....+n = n(n+1)/2}

Standard deviation :

x(i)	x(i) - mean	(x(i)-mean)^2
1	-4.5		20.25
2	-3.5		12.25
3	-2.5		6.25
4	-1.5		2.25
5	-0.5		0.25
6	0.5		0.25
7	1.5		2.25
8	2.5		6.25
9	3.5		12.25
10	4.5		20.25
		Sum =	82.5

Standard deviation :