Solving Discrete Distribution: P(X = 0 & 1)

In summary, in order to generate a random variable X, equally likely to be 0 or 1 using a biased coin, you must continuously flip the coin until you get one head and one tail, and then set X = 0 if the final flip is a head and X = 1 if the final flip is a tail. This method ensures that P{X = 0} = P{X = 1} = 1/2, as desired.
  • #1
Somefantastik
230
0
I need help getting started on this.

Want to generate a random variable X, equally likely 0, 1, using biased coin (heads probability p).

1. Flip coin, result is labled [tex]0_{1}[/tex]
2. Flip coin, result is labeled [tex]0_{2}[/tex]
3. [tex]0_{1} = 0_{2} [/tex]=> return to step 1
4. [tex]0_{2} [/tex]= heads => X = 0, [tex]0_{2}[/tex] = tails => X = 1

Show X is equally likely to be 0 or 1.

A good place to start would be showing P{X = 0} = P{X = 1}. Can you show me how to find those two probabilities?
 
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  • #2
In order to get past step 3, you have to have had one head and one tail on the two flips. Since the probs of HT and TH are equal, step 4 will give you 2 equiprobable results.
 
  • #3
Keeping in mind that this is a biased coin, can someone please show me how to explicitly find P{X = 0}? I understand that it is equal to 1/2 but I need to see how to get there.
 
  • #4
Prob(HT)=Prob(TH)=p(1-p). Prob(get to step 4) is 2p(1-p), therefore prob(X=0)=prob(X=1)=p(1-p)/(2p(1-p))=1/2.
 
  • #5
Is there a simpler way to do this where you continuously flip a coin until the last 2 results are different, that sets X = 0 if the final flip is a head, X = 1 if final flip is a tail?
 
  • #6
Somefantastik said:
Is there a simpler way to do this where you continuously flip a coin until the last 2 results are different, that sets X = 0 if the final flip is a head, X = 1 if final flip is a tail?
No. A sequence of heads followed by one tail has a different probability than a sequence of tails followed by one head. You always need to start fresh as described in your original statement.
 

Related to Solving Discrete Distribution: P(X = 0 & 1)

1. What is a discrete distribution?

A discrete distribution is a probability distribution that describes the likelihood of each possible outcome of a discrete random variable. Unlike continuous distributions, which can take on an infinite number of values, discrete distributions have a finite or countably infinite number of possible outcomes.

2. What is P(X = 0 & 1) in discrete distribution?

P(X = 0 & 1) refers to the probability that a discrete random variable X takes on the values of both 0 and 1. This can also be written as P(X = 0,1) or P(X = {0,1}). It is important to note that in probability, the "&" symbol represents the intersection of events, not the logical "and" operator.

3. How do you solve for P(X = 0 & 1)?

To solve for P(X = 0 & 1), you will need to know the probability distribution of the random variable X. This can be represented in a table, graph, or mathematical equation. Then, you can simply find the intersection of the events X = 0 and X = 1 to determine the probability of both events occurring simultaneously.

4. What is the difference between P(X = 0 & 1) and P(X = 0) + P(X = 1)?

P(X = 0 & 1) and P(X = 0) + P(X = 1) represent two different probabilities. P(X = 0 & 1) is the probability of both events X = 0 and X = 1 occurring simultaneously, while P(X = 0) + P(X = 1) is the sum of the probabilities of each event occurring individually. In some cases, these probabilities may be equal, but in other cases, they may differ.

5. How is discrete distribution used in real life?

Discrete distributions are used in many real-life situations, such as predicting the outcome of a coin toss, rolling a die, or counting the number of people with a certain characteristic in a population. They are also commonly used in statistical analysis and modeling to make predictions and decisions based on discrete data.

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