Let X1X2 be independent normal N31 N41 random variables resp

Let X1.X2 be independent normal N(3,1), N(4,1) random variables, respectively. Find P[X 1 - X2 GE 1].

Solution

X1 ~ N ( 3,1)..MEAN = 3...VARIANCE =1......

X2~ N(4,1)...MEAN =4 , VARIANCE =1......

SO, Y(let)=X1-X2 ~ N ( -1 ,2).......MEAN= 3-4 AND VARIANCE = 1+1 =2....


P( X1 - X2 >=1) = P ( Y >=1) = P [ ( Y - MEAN) / S.D >= 1-1/ ( SQRT(2) ) ] = P [ Z >= 0 ]

here z also follows normal distribution with mean = 0 and s.d =1 i.e a standdard normal distribution.

P( Z >= 0) = 1- P(Z< 0) = 0.5.....(normal distribution is symmetric)