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  Assignment 1 Stochastic Calculus 2012  † email: alex.kreinin@ca.ibm.com August 23, 2012 Problem 0.1  What is   P ( A )  if   A  is independent of itself?  Problem 0.2  Denote by   ρ  the correlation coefficient of the random variables   ξ   and   η . Prove that if   ρ  = − 1  then there exist real numbers   a <  0  and   b  such that  ξ   =  aη  +  b. Problem 0.3  Let r.v.  ξ   take only integer values with probabilities  P ( ξ   =  n ) =  cn 2 , n  = 1 , 2 ,.... Find the constant   c . Does   E ξ   exist ?  Problem 0.4  Let   ξ   and   η  be random variables taking two different values,  a  and   b . Assume that   E [ ξ  · η ] =  E [ ξ  ] · E [ η ] .Is it true that   ξ   and   η  are independent?  1  Problem 0.5 (Frequently used distributions: Binomial)  Let   0  ≤  p  ≤  1 . Consider a random variable,  ξ  , such that  P ( ξ   =  k ) =  nk   p k (1 −  p ) n − k , k  = 0 , 1 ,...,n. Find   E [ ξ  ] ,  E [ z  ξ ] ,  σ 2 ( ξ  ) . Problem 0.6 (Frequently used distributions: Geometrical)  Let   0  < p  ≤  1 . Con-sider a random variable,  η , such that  P ( η  =  k ) =  p (1 −  p ) k , k  = 1 ,...,n. Find   E [ η ] ,  E [ z  η ] ,  σ 2 ( η ) .Prove that   P ( η > i  +  j | η > i ) =  P ( η > j ) . Problem 0.7 (Frequently used distributions: Exponential)  Let   λ >  0 . Consider a random variable,  ξ  , such that  P ( ξ   ≤ x ) = 1 − exp( − λx ) , x ≥ 0 . Find   E [ ξ  ] ,  ϕ ( s ) =  E [exp( isξ  )] ,  σ 2 ( ξ  ) .Find distribution of the sum, Y  n  = n  k =1 X  k , where   X  k  form a sequence of independent random variables having the exponential distribu-tion,  Exp( λ ) . Problem 0.8  Find  P   min 1 ≤ k ≤ n ξ  k  > x  , where   ξ  k  ∼ Exp( λ ) .Find also P   max 1 ≤ k ≤ n ξ  k  ≤ x  . 2  Problem 0.9 (Convergence in distribution)  Consider a r.v.  ξ  n  = max 1 ≤ k ≤ n  X  k , where  X  k  ∼ Exp(1) . Let   η n  =  ξ n ln n . What is the limit of   η n  as   n →∞ ?  Problem 0.10  Let   ξ   and   η  be independent r.v. with   Exp( λ )  distribution. Find distribution of  β   = | η − ξ  | . Problem 0.11  Prove that  P ( | X   +  Y   | > t ) ≤ P ( | X  | > t/ 2) + P ( | Y   | > t/ 2) . Problem 0.12  Let  ϕ ( x ) = 1 √  2 πe − x 22 and  Φ( x ) =    x −∞ ϕ ( t )d t. Prove that   Φ( x )  is, indeed, cdf of a random variable. Problem 0.13  Find  P ( αX   +  βY   ≤ 3) , where   X   and   Y   are independent Normally distributed   cN  (0 , 1) -random variables and   α  and  β   are real numbers. Problem 0.14 (Cauchy distribution)  Suppose   X   = tan( ξ  ) , where   ξ   is uniformly dis-tributed on   − π 2 ,  π 2  . Find distribution of   X  . Prove that pdf of   X   is  f  X  ( x ) = 1 π (1 +  x 2 ) . Problem 0.15 (Rayleigh distribution)  Let   σ >  0 . Consider a random variable,  ξ  , such that  P ( ξ   ≤ x ) = 1 − exp  −  x 2 2 σ 2  , x >  0 . Find   E [ ξ  ] ,  Var( ξ  ) . 3  Let   X   and   Y   be independent random variables with   N  (0 ,σ 2 )  distribution. Find distribution of the random variable  Z   = √  X  2 +  Y   2 . Problem 0.16  Let   ξ   and   η  be independent uniformly distributed random variables, P ( ξ   ≤ x ) =  P ( η  ≤ x ) =  x, x >  0 . Find distribution of the r.v.  ξ  · η . Problem 0.17 (Integration by parts)  Prove that  E [ X  ] =    ∞ 0 (1 − F  X  ( x )d x −    0 −∞ F  X  ( x )d x whenever at least one of the integrals is finite. Problem 0.18  Let   X  1 ,X  2 ,...  be identically distributed nonnegative r.v. with   E X  1  <  ∞ .Prove that  X  n n  → 0  with probability   1  . Problem 0.19  Prove that the function   cos t 2 is not a characteristic function. Problem 0.20  Let   X   be a uniformly distributed random variable on the interval   [ a, b ] . Find  E e itX  . Problem 0.21  Let   ϕ k ( t )  be the characteristic function of a random variable   X  k ,  ( k  =1 , 2 ,... ) . Consider a set of positive real numbers   p 1 ,  p 2 ,  ... . Take a function   f  ( t ) = ∞  k =1  p k ϕ k ( t ) . Find conditions on   p n  such that   f   is a characteristic function. Problem 0.22  Consider a log-normal portfolio. The initial value of the portfolio is   V  0  =  S  0 .The value of the portfolio at time   t  is  V  t  =  S  0 · e ( β  − 12 σ 2 ) t + σ √  tξ , t ≥ 0 , 4
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