M378K In-Class Assignment #1

Transcription

1 The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =. Problem.. A piggy bank contains coins of three different types: T, T and T. There are twice as many type T coins as type T coins, and twice as many type T coins as type T coins. The coins are indistiguishable to touch. A coin is extracted from the piggy bank at random. Let the probability that the coin is of type T i be denoted by p i for i =,,. Find p, p and p. From the problem statement, we have that p = p = p. Since p + p + p =, we have that p = /7, p = /7 and p = /7. Problem.. Write down the definition of conditional probability. Let E, F be events such that P[F ] >. The conditional probability of E given F is defined as P[E F ] = P[E F ] P[F ] Problem.. Consider a set-up in which a transmitter is transmitting either a or a and the receiver indicates that it received either a or a. Denote the events that i =, was transmitted by T i, and the events that i =, was indicated as received by R i. It is possible to have transmission errors. In fact, you are given the following data on accuracy and the frequency of transmitted signals: P[R T ] =.99, P[R T ] =.9, and P[T ] =.75. (a) Given that the receiver indicated, what is the probability that there was an error in the transmission? (b) What is the overall probability that there was an error in transmission? () We need P[T R ]. By the Bayes formula, P[R T ]P[T ] P[T R ] = P[R T ]P[T ] + P[R T ]P[T ] (.99).75 = (.99) = =...

2 () An error will happen if T R or T R occur, i.e., P[error] = P[T R ] + P[T R ] = P[R T ] P[T ] + P[R T ] P[T ] = ( P[R T ]) P[T ] + ( P[R T ]) ( P[T ]) = = =. Problem.5. Complete the definition of independence of events below: Events A, B are said to be independent if P[A B] = P[A]P[B]. Problem.6. Two people are picked at random from a group of 5 and given $ each. After that, independently of what happened before, three people are picked from the same group - one or more people could have been picked both times - and given $ each. What is the probability that at least one person received $? Define A = {no person picked the first time was also picked the second time}, so that the probability that at least one person received $ is given by P[A c ] = P[A]. In order to compute P[A], we note that we can write A = A ij B ij, where i<j 5 A ij = { the first two people picked are i and j (not necessarily in that order)}, and B ij = { i and j are not among the next three people picked}. The sets A ij B ij and A i j B i j are mutually exclusive whenever i i or j j, so we have P[A] = P[A ij B ij ]. i<j 5 Furthermore, A ij and B ij are independent by the assumption so P[A ij B ij ] = P[A ij ]P[B ij ]. Clearly, P[A ij ] = ( 5 ), since there are ( ) 5 equally likely ways to choose people out of 5, and only one of these corresponds to the choice (i, j). Similarly, P[B ij ] = ( ), because there are ( ) 5 ways to choose people out of 5, and ( ) of those do not involve i or j. Therefore, P[A] = ( ) ) ). i<j 5 ) The terms inside the sum are all equal and there are ( ) 5 of them, so ( ) ( ) ( ) 5 P[A] = ) ) = ), and the required probability is ( ) ).

3 Problem.7. Complete the definition of a cumulative distribution function below: Let X be a random variable. The cumulative distribution function F X of X is given by: F X : R R, F X (x) = P[X x] x R. Problem.. Complete the definition of the probability (mass) function p X of a discrete random variable X. Let X be a discrete random variable. The probability (mass) function p X of X is given by p X (x) = P[X = x] for all x in the support of the random variable X. Problem.9. Seven easter eggs are hidden in a backyard. Three of the seven eggs contain a toy train from the Thomas the Tank Engine series. A toddler is on an Easter egg hunt and only really cares about the train eggs. He continues the egg hunt until he finds the first train egg. Let the random variable X represent the number of regular eggs the toddler finds before discovering the first train egg. What is the probability mass function of the random variable X? p X () = /7, p X () = (/7)(/6) = /7, p X () = (/7)(/6)(/5) = 6/5, p X () = (/7)(/6)(/5)(/) = /5, p X () = (/7)(/6)(/5)(/) = /5. Problem.. We say that a random variable X has the binomial distribution with parameters n and p if its probability (mass) function has the following form: p X (k) = ( ) n p k ( p) n k k =,,..., n. k Problem.. Complete the following definition: Let X be a discrete random variable with the probability mass function denoted by p X. The expected value E[X] of X is defined as E[X] = xp X (x) x if the series on the right-hand side is convergent absolutely, and where the sum runs through all the x in the support of X. Problem.. Let X denote the outcome of a roll of a fair, regular dodecahedron (a polyhedron with faces) with numbers,,, written on its sides. Find E[X]. E[X] = ( + + ) =. Problem.. Let X be a binomial r.v. with parameters n and p. Express E[X] in terms of n and p. E[X] = np Problem.. A random variable X is said to be continuous if

4 for any a b. there exists a density function f X : R R + such that P[a X b] = b a f X (x) dx Problem.5. Complete the following definition: Let X be a continuous random variable with the density function denoted by f X. The expected value E[X] of X is defined as if the integral is absolutely convergent. E[X] = xf X (x) dx Problem.6. Let X be a continuous random variable with the density function given as Calculate E[X]. E[X] = λ f X (x) = λe λx x >. xe λx dx = λ( x λ e λx x= + λ Problem.7. Complete the following definition: Let X be a random variable. The variance V ar[x] of X is defined as V ar[x] = E[(X E[X]) ] e λx dx) = λ. Problem.. Let X be a random variable with mean µ = and standard deviation equal to σ =. Find E[X ]. E[X ] = V ar[x] + (E[X]) = + = 5. Problem.9. Let X be a binomial r.v. with parameters n and p. Express V ar[x] in terms of n and p. V ar[x] = npq Problem.. Let X denote the number of s in throws of a fair die. Find E[X ]. Evidently, X b(, /). So, E[X ] = V ar[x] + (E[X]) = ( ) = = Problem.. We say that a random variable X is normally distributed with mean µ and variance σ if its density function equals... f X (x) = σ (x µ) π e σ x R Problem.. Let X and Y be discrete random variables on the same outcome space. The joint probability (mass) function p X,Y of the random pair (X, Y ) is defined as:

5 5 p X,Y (x, y) = P[X = x, Y = y] for all (x, y) in the support of (X, Y ). We can set the joint p.m.f. to zero for all other pairs (x, y). Problem.. Let (X, Y ) be a random pair with the joint probability (mass) function p X,Y. Then, the marginal probability (mass) function p Y of the random variable Y can be expressed in terms of p X,Y as: p X (x) = y p X,Y (x, y) for all x in the support of X and with the sum running across all y in the support of Y. Problem.. Let (X, Y ) be a random pair with the joint probability (mass) function p X,Y. Let y be a real number such that p Y (y) >. Then, the conditional probability (mass) function p X Y (,y) of the random variable X given that Y = y is defined as: for all x in the support of X. p X Y (x y) = p X,Y (x, y) p Y (y) Problem.5. A fair coin is tossed times. Let the random variable X stand for the number of heads (H) in the first two of the three coin tosses, and let Y stand for the number of tails (T) in the last two of the three coin tosses. Write down the table for the joint probability (mass) function of the random pair (X, Y ). Find the marginal distribution of Y. Determine the conditional distribution of X, given Y =. Find the distribution of the random variable Z = X + Y. Y X k P[Y = k] k P[X = k Y = ] k P[Z = k] Problem.6. Two random variables X and Y (not necessarily jointly discrete or jointly continuous!) on the same probability space are said to be independent if...

6 6 Remember that if (X, Y ) are jointly discrete, then their independence can be defined as p X,Y (x, y) = p X (x)p Y (y) for all x and y in their respective supports. Also, if (X, Y ) are jointly conitnuous, then their independence can be defined as f X,Y (x, y) = f X (x)f Y (y) for all real arguments x and y. Of course, not all random pairs are jointly discrete or jointly continuous. Luckily, there is a unified way of expressing the definition of independence and it is the following: for all pairs of numbers a and b. P[a X b, c Y d] = P[a X b]p[c Y d] Problem.7. Let X and Y be independent and both uniformly distributed on {,,..., n}. Find P[X = Y ]. P[X = Y ] = n P[X = k, Y = k] = n n = n,. k=