Given vectors \mathbf{v} = ⎡⎣−4−38⎤⎦
v=
⎣
⎢
⎡
​

−4
−3
8
​

⎦
⎥
⎤
​
, \mathbf{b_1} =
⎡⎣123⎤⎦
b
1
​
=
⎣
⎢
⎡
​

1
2
3
​

⎦
⎥
⎤
​
, \mathbf{b_2} =
⎡⎣−210⎤⎦
b
2
​
=
⎣
⎢
⎡
​

−2
1
0
​

⎦
⎥
⎤
​
and \mathbf{b_3} =
⎡⎣−3−65⎤⎦
b
3
​
=
⎣
⎢
⎡
​

−3
−6
5
​

⎦
⎥
⎤
​
all written in the standard basis, what is \mathbf{v}v in the basis defined by \mathbf{b_1}b
1
​
, \mathbf{b_2}b
2
​
and \mathbf{b_3}b
3
​
? You are given that \mathbf{b_1}b
1
​
, \mathbf{b_2}b
2
​
and \mathbf{b_3}b
3
​
are all pairwise orthogonal to each other.

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?