# Normal Distribution of… Carrots. [closed]

thank you for reading this over. I'm having trouble starting this problem...

"Suppose that the distribution of the weight of a prepackaged 1-pound bag of carrots is N($1.18, 0.07^2$) and the distribution of the weight of a prepackaged 3-pound bag of carrots is N($3.22, 0.09^2$). Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag."

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## closed as off-topic by Mike Miller, hardmath, Claude Leibovici, Graham Kemp, heropupJun 10 '14 at 5:01

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Let $X_i \sim N(1.18, 0.07^2),\ i = 1, 2, 3$ by the weights of the three prepackaged 1-pound bags, and let $Y \sim N(3.22, 0.09^2)$ by the weight of the prepackaged 3-pound bag. How do you phrase your question in terms of these? – M. Vinay Jun 10 '14 at 2:19
Please show your efforts to solve this problem, @Seraphim. You will receive better responses if it doesn't look like you're just asking people to do your homework for you. – Graham Kemp Jun 10 '14 at 4:32

Hint: Let $X_1,X_2,X_3$ be the weights of the three one pound bags, and let $Y$ be the weight of the three pound bag.

Let $W=X_1+X_2+X_3-Y$. We want the probability that $W\gt 0$.

Assume independence. Then $W$ has normal distribution, with mean $1.18+1.18+1.18- 3.22$, and variance $(0.07)^2+(0.07)^2+(0.07)^2+ (0.09)^2$.

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Sorry, I did not read your post before answering; I was just trying to review my Prob&Stats. What are the odds (ha-ha)? – user99680 Jun 10 '14 at 2:27
+1 nothing but quality from you, as always – Asimov Jun 10 '14 at 2:35
@Asimov: thanks for complimenting my jokes :). – user99680 Jun 10 '14 at 2:41
@user99680: No problem. If you want the probability, which you probably don't, it is about $0.98$. – André Nicolas Jun 10 '14 at 2:41

If the random variables $X_1, X_2, X_3, Y$ are each normal and independent, then their sum ( or linear combination ) is also normally-distributed, with mean equal to the sum of their means and variance equal to the sum of the respective variances, e.g.http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables: Then construct a new variable $Z$ from these and compute the probability that $Z \geq 0$,.

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