Sums of Random Variables


Exercises for students
Solve the following problems on paper and check your working using the above applet:

If X ~ N(10, 9), find
(i) P(X₁ + X₂ + X₃ < 15)
(ii) P(3X < 15)
(iii) P(M < 15) where M is the mean of a sample of 3 independent observations of X.

If X & Y are independent random variables such that X ~ N(10, 9) & Y ~ N(12, 16), find
(i) P(X₁ + X₂ + X₃ + Y₁ + Y₂ < 40)
(ii) P(3X + 2Y < 40)
(iii) P(X₁ + X₂ + X₃ < Y₁ + Y₂)
(iv) P(X < Y)
(v) P(3X < 2Y)
(vi) P(3X < Y₁ + Y₂)
(vii) P(X₁ + X₂ + X₃ < 2Y)
(viii) P(M < 12) where M is the mean of a sample of 4 independent observations of X.


The weight of an apple may be taken to have a normal distribution with mean 60 g and standard deviation 10 g.
The weight of an orange may be taken to have a normal distribution with mean 80 g and standard deviation 20 g.
(a) One apple is chosen at random. Find the probability that it weighs more than 45 g.
(b) Two apples are chosen at random. Find the probability that their total weight is less than 110 g.
(c) 50 apples are chosen at random. Find the probability that their mean weight is more than 59 g.
(d) One apple & one orange are chosen at random. Find the probability that the orange is more than twice as heavy as the apple.

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