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probability examples on how you can define or not define a
probability distribution on the set of natural numbers
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7. Consider an esperinent whowe sample space coansists of allive nteger(aka matural) mumbers Z+ 1,2,3,...) (i.e....
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z+...
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z+ 1,2,3, ...J (i.e. choose a random natural number) a) Can you define a probability on Z+? (b) Can you define a probability on Z+ in such a way that any two numbers are equally likely to occur? (c) Along the lines of (b), can you define a probability on the interval [0, 1] in such a way that any two numbers in this interval...
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z...
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z 1,2,3,...] (i.e. choose a random natural number) (a) Can you define a probability on Z+? (b) Can you define a probability on Z+ in such a way that any two numbers are equally likely to occur? (c) Along the lines of (b), can you define a probability on the interval [0, 1 in such a way that any two numbers in this interval are...
Consider an experiment whose sample space consists of all positive integer (a.k.a. nata) numbers Z, 1,2,3,..)...
Consider an experiment whose sample space consists of all positive integer (a.k.a. nata) numbers Z, 1,2,3,..) (i.c. choose a random natural number (a) Can you define a probability on Z? (b) Can you define a probability on Z in such a way that any two numbers are equally likely to occur? leshe in d ), con you drlune a prodability on the inerv ) 0, 1] in such a w any two numbers in this interval are cqually likely to...
4. Consider the sample space S 1,2,3,...), and assume that outcomes have the probabilities P(i)- ...
4. Consider the sample space S 1,2,3,...), and assume that outcomes have the probabilities P(i)- 2-'. For any n 2 0, define the discrete random variable Xn S0,... , n) by x,(i)-1 mod (n + 1), where mod means"modulo (a) Show that Xn converges in probability to the "identity" random variable X, defined by X(i)-. (b) Show that Xn converges in distribution to the Geom (1/2) random variable (e.g. to the time of the first Head in a sequence of...
CSCI-270 probability and statistics for computer Consider the sample space of outcomes of two throws of...
CSCI-270 probability and statistics for computer
Consider the sample space of outcomes of two throws of a fair die. Let Z = be the minimum of the two numbers that come up. List all the values of Z. Compute its probability distribution. Consider the sample space of outcomes of two tosses of a fair coin. On that space define the following random variables: X = the number of heads; Y = the number of tails on the first toss. For...
9. Consider the sample space Ω {1.2.3.4 } (the set of all natural numbers). We want...
9. Consider the sample space Ω {1.2.3.4 } (the set of all natural numbers). We want to show that there is no probability measure on 2 under which "all outcomes are equally likely" Let's argue by contradiction. Suppose P is a probability measure such that P)) has the same value for all n e2. Let's see what can go wrong. (a) Suppose P)> 0. Which axiom of probability will be violated? (b) Suppose P((n)) = 0, which axiom of probability...
4. Consider a inap φ : I 1,11 > 10, 1] defined by φ(z) :-12. Let...
4. Consider a inap φ : I 1,11 > 10, 1] defined by φ(z) :-12. Let X and Y be random variables related by the map φ, i.c., Y-o(X) (their sample spaces are then given by SX-1 1,11 and SY-10,1]). Let FY be the σ-algebra and Hy the probability measure you worked out in problem 3. Compute the adaptod ơ algebra X and the corresponding probability measure x (i.e., use the formula X (ф ія, )-, Y (S.) for any...
5. Consider the sample space Ω = [0, 1]. Let P be a probability function such...
5. Consider the sample space Ω = [0, 1]. Let P be a probability function such that for any interval fa, b, P(a, b-b-a. In other words, probabilty of any interval is its length Let us start with Co [0, 1, and at nth step, we define Cn by removing an interval of length 1/3 from the middle of each interval in Cn-1 For example, C1-[0, 1/3 u [2/3,1], C2-[0,1/9)U[2/9,1/3 U [2/3,7/9 U[8/9, 1] and so on. Here is a...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any interval [a, b], P([a, b) b- a. In other words, probabilty of any interval is its length. Let us start with Co 10, 1], and at nth step, we define C, by removing an interval of length 1/3° from the middle of each interval in Cn-1. For example, G = [0, 1/3ju [2/3, 11, c2 [0, 1/9] U [2/9, 1/3] U [2/3,7/9] U [8/9,...
Consider the Markov chain on state space {1,2, 3,4, 5, 6}
Consider the Markov chain on state space {1,2, 3,4, 5, 6}. From 1 it goes to 2 or 3 equally likely. From 2 it goes back to 2. From 3 it goes to 1, 2, or 4 equally likely. From 4 the chain goes to 5 or 6 equally likely. From 5 it goes to 4 or 6 equally likely. From 6 it goes straight to 5. (a) What are the communicating classes? Which are recurrent and which are transient? What...
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z+ 1,2,3, ...J (i.e. choose a random natural number) a) Can you define a probability on Z+? (b) Can you define a probability on Z+ in such a way that any two numbers are equally likely to occur? (c) Along the lines of (b), can you define a probability on the interval [0, 1] in such a way that any two numbers in this interval...
7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z 1,2,3,...] (i.e. choose a random natural number) (a) Can you define a probability on Z+? (b) Can you define a probability on Z+ in such a way that any two numbers are equally likely to occur? (c) Along the lines of (b), can you define a probability on the interval [0, 1 in such a way that any two numbers in this interval are...
Consider an experiment whose sample space consists of all positive integer (a.k.a. nata) numbers Z, 1,2,3,..) (i.c. choose a random natural number (a) Can you define a probability on Z? (b) Can you define a probability on Z in such a way that any two numbers are equally likely to occur? leshe in d ), con you drlune a prodability on the inerv ) 0, 1] in such a w any two numbers in this interval are cqually likely to...
4. Consider the sample space S 1,2,3,...), and assume that outcomes have the probabilities P(i)- 2-'. For any n 2 0, define the discrete random variable Xn S0,... , n) by x,(i)-1 mod (n + 1), where mod means"modulo (a) Show that Xn converges in probability to the "identity" random variable X, defined by X(i)-. (b) Show that Xn converges in distribution to the Geom (1/2) random variable (e.g. to the time of the first Head in a sequence of...
CSCI-270 probability and statistics for computer
Consider the sample space of outcomes of two throws of a fair die. Let Z = be the minimum of the two numbers that come up. List all the values of Z. Compute its probability distribution. Consider the sample space of outcomes of two tosses of a fair coin. On that space define the following random variables: X = the number of heads; Y = the number of tails on the first toss. For...
9. Consider the sample space Ω {1.2.3.4 } (the set of all natural numbers). We want to show that there is no probability measure on 2 under which "all outcomes are equally likely" Let's argue by contradiction. Suppose P is a probability measure such that P)) has the same value for all n e2. Let's see what can go wrong. (a) Suppose P)> 0. Which axiom of probability will be violated? (b) Suppose P((n)) = 0, which axiom of probability...
4. Consider a inap φ : I 1,11 > 10, 1] defined by φ(z) :-12. Let X and Y be random variables related by the map φ, i.c., Y-o(X) (their sample spaces are then given by SX-1 1,11 and SY-10,1]). Let FY be the σ-algebra and Hy the probability measure you worked out in problem 3. Compute the adaptod ơ algebra X and the corresponding probability measure x (i.e., use the formula X (ф ія, )-, Y (S.) for any...
5. Consider the sample space Ω = [0, 1]. Let P be a probability function such that for any interval fa, b, P(a, b-b-a. In other words, probabilty of any interval is its length Let us start with Co [0, 1, and at nth step, we define Cn by removing an interval of length 1/3 from the middle of each interval in Cn-1 For example, C1-[0, 1/3 u [2/3,1], C2-[0,1/9)U[2/9,1/3 U [2/3,7/9 U[8/9, 1] and so on. Here is a...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any interval [a, b], P([a, b) b- a. In other words, probabilty of any interval is its length. Let us start with Co 10, 1], and at nth step, we define C, by removing an interval of length 1/3° from the middle of each interval in Cn-1. For example, G = [0, 1/3ju [2/3, 11, c2 [0, 1/9] U [2/9, 1/3] U [2/3,7/9] U [8/9,...
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