You are a scientist. You try to grow big tomatoes. You have two varieties, calle

Question

You are a scientist. You try to grow big tomatoes. You have two varieties, called Awesome Red and Big Boy. The weight of an Awesome Red you think is normally distributed with mean 0.4kg and standard deviation 0.2kg. The weight of a Big Boy you think is normally distributed with mean 0.8kg and standard deviation 0.2kg. Assuming these weights are independent, what's the probability that if you pick a pair consisting of an Awesome Red and a Big Boy tomato, that the weight of the Big Boy is twice that of the Awesome Red?

Explanation / Answer

let X denotes the weight of awesome red in kg

Y denotes the weight of big boy in kg

so X~N(0.4,0.22) and Y~N(0.8,0.22) independently

now we need to find the probability that weight of Big boy is twice that of Awesome Red

so P[Y>2X]=P[Y-2X>0]

now E[Y-2X]=E[Y]-E[2X]=0.8-0.4*2=0

V[Y-2X]=V[Y]+4*V[X]=0.22+4*0.22=0.2 [since X and Y are independent, covariance is 0]

so Y-2X being a linear combination of normal variates , it also follows normal distribution

so Z=Y-2X~N(0,0.2)

so P[Y-2X>0]=P[Z>0]=0.5 since normal distribution is symmetric about its mean

hence the required probability is 0.5 [answer]

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