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Introduction to Probability Theory II

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Introduction to Probability Theory II

Exercise 6, Autumn 2009

1. Buffon’s needle problem Let the length of a needle be 2k units, where 0 < k < a. The needle is dropped on a paper ruled with parallel lines 2a units apart. Calculate the probability that the needle intersects one of the lines.

Hint: LetX be the distance from the centre of the needle to the closest line and Y the acute angle between the needle and the lines. Now X and Y are independent. Furthermore X is uniformly distributed on the interval]0,a[ andY is uniformly distributed on the interval ]0, π[.

2. Assume that the signal arriving from a satellite is S=X+Y, where X is result of an observation and Y is the interference. Assume that X y Y, X∼N(µ, σ21) and (Y∼N(0, σ22)). Find

a) Corr(S,X),

b) Distribution of X conditional on S=s.

(Hint: Present the random vector (X,S) as an affine transformation of the random vector (U,V), where (U,V) ∼ N(0,I). Use this to find the density function of random vector (X,S)).

3. Let the density function of random vector (X,Y) be f, where f(x,y)=ce−x2−2y2 for every (x,y)∈R2, with c>0.

a) Determinec.

b) What is this distribution called?

c) FindE(X), E(Y)and Corr(X,Y).

4. Let the density function of the random vector (X,Y) be f, when f(x,y)=ce−x2−2y2 kaikilla (x,y)∈R2,

where c is a constant.

a) Determine the value of c.

b) Name the distribution.

c) FindE(X), E(Y)and Corr(X,Y).

d) Assume that the distribution of random vector(X,Y)is a2–dimensional normal distribution with density function

f(x,y)= 1 2√

2πexph

1/8(3x2+2xy+3y2−14x−10y+19)i

for every (x,y)∈R2.

Find the expected values forX and Y, and their covariance matrix.

Hint: Find vector z0 and a matrix C such that the argument of exponential function is−1/2(z−z0)TC−1(z−z0), ifz=

"

x y

#

andz0=

"

x0

y0

# . 5. Two points are placed on a line segment of length a randomly and inde-

pendent of each other.

a) Let 0 < x < a. Calculate the probability that the distance between points is greater than x.

b) Calculate expected valua of the distance.

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