Solution of Homework 1

Problem 1

Let the rv X have the probability mass function f(x;λ) = c(λ+x), x=1,2,..,N, where N is known, c is a constant and λ is an unknown parameter.

  1. find c
  2. find E[X]

Problem 2

Let the rv (X,Y) be the uniform distribution, on the area \(0<x<y^2<1, 0<y<1\).

  1. find cor(X,Y)

  1. find E[X|Y=y]

Problem 3

Say we have discrete random variables \(X_1,..,X_{50}\), independent, with \(P(X_i=x)=c(1+(x-2)^2)\), \(x=0,1,2,3,4\). Find (an approximate value of)

\[P(X_1+...+X_{50}>110)\]

We will use the central limit theorem, so we need

x=0:4
cc=sum(1+(x-2)^2)
ex=sum(x*(1+(x-2)^2))/cc
ex2=sum(x^2*(1+(x-2)^2))/cc
c(cc, ex, ex2, ex2-ex^2, sqrt(ex2-ex^2))
## [1] 15.000000  2.000000  6.933333  2.933333  1.712698

\[ \begin{aligned} &P(X_1+...+X_{50}>110) = \\ &P(\bar{X}>110/50) = \\ &P\left(\sqrt{n}\frac{\bar{X}-\mu}{\sigma}>\sqrt{50}\frac{110/50-2}{1.713}\right) = \\ &1-\Phi(0.826) =0.194 \end{aligned} \]

Problem 4

Let \(X\sim Exp(1)\) and \(Y\sim Exp(2)\), independent. Find \(P(X+Y<1)\).

We will need the density of \(X+Y\). By the formula derived in (2.1.3) we have

\[ \begin{aligned} &f_{X+Y}(z) =\int_{-\infty}^\infty f_{X}(z-x)f_Y(x) dx= \\ &\int_0^z e^{-(z-x)}2e^{-2x}dx =\\ &\int_0^z 2e^{-z-x}dx =\\ &-2e^{-z-x}\vert_0^z=2e^{-z}-2e^{-2z} \end{aligned} \] and so

\[ \begin{aligned} &P(X+Y<1) = \\ &\int_0^1 2e^{-z}-2e^{-2z} dz = \\ &-2e^{-z}+e^{-2z}\vert_0^1 = \\ &-2e^{-1}+e^{-2} +2-1 = 0.4 \end{aligned} \]