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Suppose that a radio contains six transistors, two of which are defective.
Three transistors are selected at random, removed from the radio, and
inspected. Let Y equal the number of defectives observed,where Y =
0, 1, or 2. Find the probability distribution for Y . Express your results
graphically as a probability histogram.
Suppose that X has probability density function f (x) = 2x between 0 and
1 and f (x) = 0 elsewhere. Compute V ar(X) and V ar(X
2
).
Suppose that Y possesses the density function
f (y) =
cy, 0 ≤ y ≤ 2
0, elsewhere
(a) Find the value of c that makes f (y) a probability density function.
(b) Find F (y).
(c) Use F (y) to find P (1 ≤ Y ≤ 2).
(d) Use f (y) and geometry to find P (1 ≤ Y ≤ 2).
The volume in a set of wine bottles is known to follow a N (µ, 25) distribution. You take a sample of the bottles and measure their volumes.How
many bottles do you have to sample to have a 95% confidence interval for
µ with width 1?
Let Y1 and Y2 have joint density function:
f (y1
, y2) =
e−(y 1 +y 2 ) , y1 > 0, y2 > 0,
0, elsewhere
what is:
(a) P (Y1 < 1, Y2 > 5)?
(b) P (Y1 + Y2 < 3)?
Denote
S
0
2 =
P n
i=1 (Yi − Y¯ )2
n
and S2 =
P n
i=1 (Yi − Y¯ )2
n − 1
.
If the Yi are i.i.d. and normally distributed,
(a) find V (S
0
2
).
(b) show that V (S
2
) > V (S
0
2
).
(c) Therefore, which estimator do you prefer? Why?