text: math
  
  A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K= 5. For k = 1,2,...,K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xkare independent of one another and of the coin flips. Let X=sum(Xk). Find the mean and variance of X.
  
  1. 👍 1
  2. 👎 0
  3. 👁 1,021
  
  1. 👍 0
  2. 👎 0
  2. Firstly, I assume N=K in your solutions. The expected value and variance of X can be found via Law of Iterated Expectation (LIE) and Law of Total Variance (LTV):
  
  E[X]=E[E[X|K]], var(X)=E[var(X|K)]+var(E[X|K])
  
  For the expectation, your approach is correct, but it can be found via LIE:
  E[X|K]=KE[Xk]→E[KE[Xk]]=E[K]E[Xk]
  You just need to correct your expectation for K: E[K]=1/p+1, since it is of the form 1+Y, where Y is a geometric RV with parameter p. Also, note that var(K)=var(1+Y)=var(Y)=(1−p)/p2 as yours.
  
  For the variance, we need var(X|K)=var(∑Xk|K)=Kvar(Xk), and by LTV:
  var(X)=E[Kvar(Xk)]+var(KE[Xk])=var(Xk)E[K]+E[Xk]2var(K)
  
  Substituting:
  
  E[X] = 15/2
  Var[X] = 18.75
  
  1. 👍 3
  2. 👎 0
  
  Similar Questions
  
  1. Probability
  
  Problem 1 Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X2]=E[Y2]=E[Z2]=5. Find cov(XY,XZ). cov(XY,XZ)= ? Problem 2. Let X be a standard normal random variable. Another random variable is determined as
  
  2. Probability
  
  We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. We assume that conditioned on Q=q, all coin tosses are independent. Let
  
  3. Probability
  
  We have a red coin, for which P(Heads)=0.4, a green coin, for which P(Heads)=0.5 and a yellow coin for which P(Heads)=0.6. The flips of the same or of different coins are independent. For each of the following situations,
  
  4. Mathematics
  
  Let X be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from X). In case of Heads, we let Y=X. In case of Tails, we let Y=−X.
  
  1. Math
  
  Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and
  
  2. Probability
  
  A defective coin minting machine produces coins whose probability of Heads is a random variable Q with PDF fQ(q)={3q2,0,if q∈[0,1],otherwise. A coin produced by this machine is tossed repeatedly, with successive tosses assumed
  
  3. Statistics
  
  Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 . Find cov(XY,XZ) . cov(XY,XZ)= Let X be a standard normal random variable. Another random variable is determined as follows. We flip a
  
  4. Probability
  
  Let Θ be a continuous random variable that represents the unknown bias (i.e., the probability of Heads) of a coin. a) The prior PDF fΘ for the bias of a coin is of the form fΘ(θ)=aθ9(1−θ), for θ∈[0,1], where a is a
  
  1. Math
  
  A defective coin minting machine produces coins whose probability of Heads is a random variable Q with PDF fQ(q)={5q4,0,if q∈[0,1],otherwise. A coin produced by this machine is tossed repeatedly, with successive tosses assumed
  
  2. Probability
  
  Question:A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5. For K=1,2,3...K,
  
  3. Probability
  
  Alice has two coins. The probability of Heads for the first coin is 1/3, and the probability of Heads for the second is 2/3. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and
  
  4. probability
  
  A fair coin is flipped independently until the first Heads is observed. Let K be the number of Tails observed before the first Heads (note that K is a random variable). For k=0,1,2,…,K, let Xk be a continuous random variable
explanation: The document describes a math problem involving a random variable K, which is the number of coin tosses until the first Heads is observed plus 1, and a set of continuous random variables Xk that are uniform over the interval [0, 5]. The problem asks to find the mean and variance of the sum of these random variables, X. The document also provides a solution using the Law of Iterated Expectation (LIE) and the Law of Total Variance (LTV).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: math
  
  A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K= 5. For k = 1,2,...,K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xkare independent of one another and of the coin flips. Let X=sum(Xk). Find the mean and variance of X.
  
  1. 👍 1
  2. 👎 0
  3. 👁 1,021
  
  1. 👍 0
  2. 👎 0
  2. Firstly, I assume N=K in your solutions. The expected value and variance of X can be found via Law of Iterated Expectation (LIE) and Law of Total Variance (LTV):
  
  E[X]=E[E[X|K]], var(X)=E[var(X|K)]+var(E[X|K])
  
  For the expectation, your approach is correct, but it can be found via LIE:
  E[X|K]=KE[Xk]→E[KE[Xk]]=E[K]E[Xk]
  You just need to correct your expectation for K: E[K]=1/p+1, since it is of the form 1+Y, where Y is a geometric RV with parameter p. Also, note that var(K)=var(1+Y)=var(Y)=(1−p)/p2 as yours.
  
  For the variance, we need var(X|K)=var(∑Xk|K)=Kvar(Xk), and by LTV:
  var(X)=E[Kvar(Xk)]+var(KE[Xk])=var(Xk)E[K]+E[Xk]2var(K)
  
  Substituting:
  
  E[X] = 15/2
  Var[X] = 18.75
  
  1. 👍 3
  2. 👎 0
  
  Similar Questions
  
  1. Probability
  
  Problem 1 Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X2]=E[Y2]=E[Z2]=5. Find cov(XY,XZ). cov(XY,XZ)= ? Problem 2. Let X be a standard normal random variable. Another random variable is determined as
  
  2. Probability
  
  We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. We assume that conditioned on Q=q, all coin tosses are independent. Let
  
  3. Probability
  
  We have a red coin, for which P(Heads)=0.4, a green coin, for which P(Heads)=0.5 and a yellow coin for which P(Heads)=0.6. The flips of the same or of different coins are independent. For each of the following situations,
  
  4. Mathematics
  
  Let X be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from X). In case of Heads, we let Y=X. In case of Tails, we let Y=−X.
  
  1. Math
  
  Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and
  
  2. Probability
  
  A defective coin minting machine produces coins whose probability of Heads is a random variable Q with PDF fQ(q)={3q2,0,if q∈[0,1],otherwise. A coin produced by this machine is tossed repeatedly, with successive tosses assumed
  
  3. Statistics
  
  Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 . Find cov(XY,XZ) . cov(XY,XZ)= Let X be a standard normal random variable. Another random variable is determined as follows. We flip a
  
  4. Probability
  
  Let Θ be a continuous random variable that represents the unknown bias (i.e., the probability of Heads) of a coin. a) The prior PDF fΘ for the bias of a coin is of the form fΘ(θ)=aθ9(1−θ), for θ∈[0,1], where a is a
  
  1. Math
  
  A defective coin minting machine produces coins whose probability of Heads is a random variable Q with PDF fQ(q)={5q4,0,if q∈[0,1],otherwise. A coin produced by this machine is tossed repeatedly, with successive tosses assumed
  
  2. Probability
  
  Question:A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5. For K=1,2,3...K,
  
  3. Probability
  
  Alice has two coins. The probability of Heads for the first coin is 1/3, and the probability of Heads for the second is 2/3. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and
  
  4. probability
  
  A fair coin is flipped independently until the first Heads is observed. Let K be the number of Tails observed before the first Heads (note that K is a random variable). For k=0,1,2,…,K, let Xk be a continuous random variable
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document describes a math problem involving a random variable K, which is the number of coin tosses until the first Heads is observed plus 1, and a set of continuous random variables Xk that are uniform over the interval [0, 5]. The problem asks to find the mean and variance of the sum of these random variables, X. The document also provides a solution using the Law of Iterated Expectation (LIE) and the Law of Total Variance (LTV).
  
  Label: question-and-answer