text: # Tagged Questions
  
  This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...
  
  22 views
  
  ### Finding PDF of function of a random variable
  
  Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
  46 views
  
  ### Expected Value help
  
  A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. What is E[X]? To be honest, I don't get this. The answer was ...
  21 views
  
  ### Special dice generating non-decreasing sequence
  
  Suppose that, when rolled for the first time, a special 6-sided dice shows $1,\ldots, 6$ with probability $\frac{1}{6}$ each, and then, upon rerolling, shows with equal probability a number greater or ...
  45 views
  
  ### Probability of number of people who know a rumor
  
  Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
  37 views
  
  ### Central Limit Theorem and Mean time between failures
  
  I was reading up about RAID, and the text said: Suppose that the mean time to failure of a single disk is $100000$ hours. Then the mean time to failure of some disk in an array of 100 disks ...
  51 views
  
  ### The expected value of the smallest number in sample $S$ is:
  
  We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
  9 views
  
  36 views
  
  ### what is the CDF of $f(x)=\frac{3x^2}{2}$?
  
  This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...
  18 views
  
  ### Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?
  
  Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
  385 views
  
  ### Finding covariance from marginal densities.
  
  A quarter is bent so that the probabilities of heads and tails are 0.40 and 0.60. If it is tossed twice, what is the covariance of Z, the number of heads obtained on the first toss, and W, the total ...
  11 views
  
  ### Is the mixture of Exponential family distributions an Exponential family distribution too?
  
  Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
  49 views
  
  ### What is right solution for this probability problem?
  
  This drug can cure $90$% of all diseases. What is probabilty of successful healing at least $18$ people of $20$ people, who have taken the drug? What is the right solution and why? From my point of ...
  25 views
  
  172 views
  
  ### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
  
  In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
  15 views
  
  ### coincidence of recurrent random processes with infinite expected periods
  
  That subject might not be quite accurate, but let me clarify. At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, ...
  ### For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.
  Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
  a fair die is rolled 100 times. What is the probability that "6" appears more than 15 times? Use the normal approximation with continuity correction. I've found the mean to be $100/6$ or $50/3$ and ...
explanation: The document is a collection of probability questions from a math forum or textbook. Each question deals with different aspects of probability theory, such as finding the probability density function (PDF) of a transformed random variable, calculating expected values, and understanding the behavior of random processes. The document contains multiple questions, but no answers are provided.
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: # Tagged Questions
  
  This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...
  
  22 views
  
  ### Finding PDF of function of a random variable
  
  Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
  46 views
  
  ### Expected Value help
  
  A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. What is E[X]? To be honest, I don't get this. The answer was ...
  21 views
  
  ### Special dice generating non-decreasing sequence
  
  Suppose that, when rolled for the first time, a special 6-sided dice shows $1,\ldots, 6$ with probability $\frac{1}{6}$ each, and then, upon rerolling, shows with equal probability a number greater or ...
  45 views
  
  ### Probability of number of people who know a rumor
  
  Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
  37 views
  
  ### Central Limit Theorem and Mean time between failures
  
  I was reading up about RAID, and the text said: Suppose that the mean time to failure of a single disk is $100000$ hours. Then the mean time to failure of some disk in an array of 100 disks ...
  51 views
  
  ### The expected value of the smallest number in sample $S$ is:
  
  We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
  9 views
  
  36 views
  
  ### what is the CDF of $f(x)=\frac{3x^2}{2}$?
  
  This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...
  18 views
  
  ### Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?
  
  Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
  385 views
  
  ### Finding covariance from marginal densities.
  
  A quarter is bent so that the probabilities of heads and tails are 0.40 and 0.60. If it is tossed twice, what is the covariance of Z, the number of heads obtained on the first toss, and W, the total ...
  11 views
  
  ### Is the mixture of Exponential family distributions an Exponential family distribution too?
  
  Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
  49 views
  
  ### What is right solution for this probability problem?
  
  This drug can cure $90$% of all diseases. What is probabilty of successful healing at least $18$ people of $20$ people, who have taken the drug? What is the right solution and why? From my point of ...
  25 views
  
  172 views
  
  ### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
  
  In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
  15 views
  
  ### coincidence of recurrent random processes with infinite expected periods
  
  That subject might not be quite accurate, but let me clarify. At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, ...
  ### For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.
  Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
  a fair die is rolled 100 times. What is the probability that "6" appears more than 15 times? Use the normal approximation with continuity correction. I've found the mean to be $100/6$ or $50/3$ and ...
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a collection of probability questions from a math forum or textbook. Each question deals with different aspects of probability theory, such as finding the probability density function (PDF) of a transformed random variable, calculating expected values, and understanding the behavior of random processes. The document contains multiple questions, but no answers are provided.
  
  Label: multiple-question