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  STAT 333: Assignment 1 Due : October 9th in class, or in Pengfei or Martin’s office no later than 5pm on October 10th. Note : Please use the cover page for the assignment.1. A collection of   n  coins is flipped. The outcomes are independent and the  i th coin comesup heads with probability  α i ,  i  = 1 ,...,n . Suppose  α 1  = 1 / 2. Find the probabilitythat the total number of heads to appear in the  n  coins will be an even number.2. Suppose n letters are randomly selected with replacement from the set S   =  { A,B,C,...I  } (9 letters in total). Let  X   denote the number of distinct letters which appear in the  n picks. For example, if   n  = 5 and the selected 5 letters are “ACEGA”, then  X   = 4.Calculate  E  ( X  ) and Var( X  ).3. A sequence of independent digits (“0” or “1”) is generated, where for each digit theprobability of generating a “1” is  p , 0  < p <  1. Let  T   be the waiting time (number of steps in the sequence) until we first observe “010”.(a) Find  E  ( T  ).(b) Find the value of   p  where  E  ( T  ) is minimized. Note:  You may use a computer to calculate this value.4. A mouse is at the center of a maze. Three doors lead out of the maze. Door 1 leadsback to the center after 7 minutes of scampering. Door 2 leads back to the center after6 minutes of scampering. Door 3, however, after 4 minutes of scampering, splits intotwo tunnels, Tunnel  A  and Tunnel  B . If the mouse chooses Tunnel  A , it gets out of the maze after 1 minutes. If the mouse chooses Tunnel  B , it gets out of the maze after3 minutes. This information is displayed in the diagram below.Door 1 Tunnel  A 1 min         ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ Center        7 mins      ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑        6 mins                4 mins        Door 3         ExitDoor 2 Tunnel  B 3 mins                   1  (a) Let  X   be number of minutes for the mouse to get out of the maze. Suppose themouse always chooses at random between any door (or tunnel) at each opportu-nity. Find  E  ( X  ) and Var( X  ).(b) Suppose that the mouse always takes a rest before choosing a door (but not thetunnel) – even for the first door it picks. Assume that each rest time beforechoosing a door is Uniformly distributed between 0 and 2 minutes, independentlyof anything else. Let  Y   be the total number of minutes that the mouse spentresting before coming out of the maze. Find  E  ( Y  ) and Var( Y  ).5. Let  X  1 ,...,X  n  be independent Exponential random variables each with rate  λ i ,  i  =1 ,...,n . (Note that  X  1 ,...,X  n  are not necessarily identically distributed.)(a) Find the probability density function of   Y   = min { X  1 ,X  2 ,...,X  n } . What type of distribution is the random variable  Y  ?(b) Find  P  ( X  1  < X  2 ) or equivalently  P  ( X  1  = min { X  1 ,X  2 } ).(c) Find  P  ( X  1  <  min { X  2 ,...,X  n } ) or equivalently  P  ( X  1  = min { X  1 ,X  2 ,...,X  n } ). Hint:  Use the results in Part (a) and Part (b).6. Suppose  X  1 ,X  2 ,...  are independent and identically distributed Uniform random vari-ables over [0 , 1]. Assume that  N   ∼  Geo(1 / 2) independently of   X  1 ,X  2 ,... .Let  Y   =  N i =1 X  i .(a) Find  E  ( Y  ).(b) Find  E  ( Y  2 ).(c) Find Var( Y  ).2
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