A mouse walks in a maze that is an orthogonal grid made of corridors that intersect at crossings one foot apart, stopping at eve
ry intersection. Using coordinates (with units in feet), it starts at the origin, then moves equally likely up to (0, 1) or down to (0, −1) or right to (1, 0) or left to (−1, 0) by one unit until the next crossing. Then it stops, picks another random direction (up, down, right or left) equally likely and moves by another unit till the next crossing. Every time it stops, its position is a vector (a, b) where both a and b are integers. Scientists let the mouse walk n feet, after which its position is (X, Y ) and it is at distance D from the origin (D = √ X2 + Y 2).
a) What is Cov(X,Y)?
b) Are X and Y independent?
c) What is E(DP)?
a) What is Cov(X,Y)?
b) Are X and Y independent?
c) What is E(DP)?
1 answer:
You might be interested in
Answer:
Therefore the required polynomial is
M(x)=0.83(x³+4x²+16x+64)
Step-by-step explanation:
Given that M is a polynomial of degree 3.
So, it has three zeros.
Let the polynomial be
M(x) =a(x-p)(x-q)(x-r)
The two zeros of the polynomial are -4 and 4i.
Since 4i is a complex number. Then the conjugate of 4i is also a zero of the polynomial i.e -4i.
Then,
M(x)= a{x-(-4)}(x-4i){x-(-4i)}
=a(x+4)(x-4i)(x+4i)
=a(x+4){x²-(4i)²} [ applying the formula (a+b)(a-b)=a²-b²]
=a(x+4)(x²-16i²)
=a(x+4)(x²+16) [∵i² = -1]
=a(x³+4x²+16x+64)
Again given that M(0)= 53.12 . Putting x=0 in the polynomial
53.12 =a(0+4.0+16.0+64)
=0.83
Therefore the required polynomial is
M(x)=0.83(x³+4x²+16x+64)