Q.1. i) In five tests, one student averages 63.2 with a standard deviation of 3.3, whereas another student averaged 78.8 with a standard deviation of 5.3. Which student is relatively more consistent?
ii) An agricultural cooperative claims that 90% of the watermelons shipped out are ripe and ready to eat. Find the probabilities that among 18 watermelons shipped out:
a. All 18 are ripe and ready to eat.
b. At least 16 are ripe and ready to eat.
Q.2. Three events A, B, and C are pairwise independent if each pair is independent. They are mutually independent if they are pairwise independent and in addition P(A ∩ B ∩ C) = P(A)P(B)P(C). Suppose we roll two 6-sided dice. Consider the events:
A = 'odd on die 1'
B = 'odd on die 2'
C = 'odd sum'
Are A, B, and C pairwise independent? Are they mutually independent?
Q.3. Suppose X and Y have joint pdf f(x,y) = c(x^2 + xy) on [0,1] x [0,1].
i) Find c and joint cdf F(x,y)
ii) Find the marginal cumulative distribution functions Fx and Fy and the marginal pdf fX and fY.
iii) Find E(X) and Var(X)
Q.4. The daily number of orders filled by the parts department of a repair shop is a random variable with μ = 142 and σ = 12. According to Chebyshev's theorem, with what probability can we assert that on any one day, it will fill between 82 and 202 orders?
Q.5. A random sample of size 25 from a normal population has the mean x̄ = 47.5 and the standard deviation s = 8.4. Does this information tend to support or refute the claim that the mean of the population is μ = 42.1?