L8 02 03 Geometric Random Variables

What is the purpose of simulating a fair coin using a biased coin?

To create a fair outcome (heads or tails) from a biased source, ensuring equal probabilities.

How can you simulate a fair coin using a biased coin?

Flip the biased coin twice and use the outcomes "heads-tail" and "tails-head" to represent heads and tails of the fair coin.

What outcomes from flipping a biased coin require a redo?

The outcomes "heads-heads" and "tails-tails" require a redo.

What is the expected number of flips needed to simulate a fair coin from a biased coin?

The expected number of flips is two.

How can you simulate a biased coin using a fair coin?

Generate a random number between 0 and 1, and use the segments corresponding to the biased coin's probabilities to determine the outcome.

What does the segment corresponding to heads represent when simulating a biased coin?

It represent the probability of heads, which in this case is one-seventh.

What is the significance of generating binary digits when simulating a biased coin?

The binary digits help determine which segment of the unit interval the random number falls into, indicating the outcome.

Why is the expected number of flips to simulate a biased coin only two?

Each flip has a 50% chance of leading to a conclusive outcome, allowing for termination after two flips on average.

How can the discussed methods be generalized beyond coins?

They can be applied to sample from multiple objects with different probabilities, such as vowels in the English Language.

What is "roulette wheel selection"?

A method of sampling where the width of each sector on a wheel corresponds to the probabilty of each object being selected.

How does binary searching relate to sampling from a set of N possibilites?

Binary searching can efficiently determine which segment a random number falls into, allwoing for quick sampling.

What is the relationship between the discussed algorithms and data compression?

The algorithm can be used to assign variable-length codes to frequently occuring elements, optimizing space in data represenattion.

What is the role of the unit interval [0,1] in simulating a biased coin?

The unit interval is divided into segments that correspond to the probabilites of heads and tails, helping to determine the outcome based on where a random number falls.

How does the concept of ambiguity play a role in the coin flipping process?

Ambiguity arises when the outcome of a flip does not clearly indicate heads or tails, requiring additional flips until a conclusive result in achieved.

What is the significance of the binary tree analogy in generating random numbers?

The binary tree represents the paths taken during coin flips, where each flip determines the direction (left for heads, right for tails) and helps visualize the generation of binary digits.

How can the process of generating binary digits be stopped?

The process stops when a generated digit differs from the corresponding digit in the target probability (ex. One-seventh) indicating whether the random number is greater or less than that value.

What is arithmetic coding in the context of data compression?

Arithmetic coding is a technique that assigns variable-length binary codes to symbols based on their probabilities, allowing for efficient data representations.

How does the concept of variable-length coding relate to the frequency of elements?

More frequently occurring elements are assigned shorter codes, while less frequent elements receive longer codes, optimizing the overall space used in encoding.

What is the expected outcome when repeatedly sampling from a biased distribution?

The expected outcome aligns with the defined probabilities of the distribution, ensuring that the sampling reflects the intended bias.

How can preprocessing improve the efficiency of sampling from distribution?

By organizing the segments of the distribution in a sorted array, binary search can be used to quickly determine which segment a random number falls into, enhancing sampling speed.

What is the expected cover time in a network of machines?

It is expected number of steps required for a message to be received by every machine in the network.

How is the expected cover time analyzed?

It is analyzed using geometric random variables that correspond to the time taken in each phase of the message spreading process.

What happens to the expected number of steps as more machines receive the message?

The expected number of steps increases, as it becomes less likely to reach the remaining machines.

What is the probability of success in the last phase when N-1 machines have heard the message?

The probability of success is 1/N, leading to an expected number of trials equal to N.

How is the total expected number of steps calculated?

By summing the expected steps for each phase, resulting in N times a harmonic series, which approximates to N log N.

What is the randomized reduction lemma?

It states that if an algorithm reduces the problem size by half with a certain probability, the expected number of steps needed is logarithmic.

How many steps are expected before the first successful reduction occurs?

At most four steps are expected before the first reduction, given a probability of at least 1/4.

How many reduction are needed to terminate the algorithm?

Log base two of N reductions are needed to reach the base case.

What is the upper bound for the total number of steps needed by the algorithm?

It is bounded by the sum of expected steps leading up to each reduction phase.

What is the significance of geometric random variables in this context?

They help in modeling the expected number of trials until success in each phase of the algorithm.