Another probability - random variables, generating functions

In summary, when reading a book, the probability of detecting a mistake on a page is independent of other mistakes and can be represented by a Poisson RV with parameter a. The expected number of pages with no mistakes is 1 - e^-a. To find P[D=k|M=m], the probability of detecting k mistakes given m total mistakes on a page, we can use the formula P[D=k|M=m] = P(D=k union M=m)/P(M=m). To find P[D=k], the probability of detecting k mistakes on a page regardless of the total number of mistakes, we can use the formula P[D=k] = P(D=k union M=m) + P(D=k union M=m+1) + ... However
  • #1
Kate2010
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Homework Statement


When reading a book, you detect each mistake with probability p, independent of other mistakes. Let M denote the amount of mistakes on a certain page and D be the number that you detect on that page. Write down P[D=k|M=m] and find for k>=0 P[D=k].


Homework Equations


P(x=r) = (a^r)(e^-a)/(r!), poisson


The Attempt at a Solution


Earlier in the question I worked out for a textbook with n pages, number of mistakes on each page is poisson RV with parameter a, independent of mistakes on all other pages, that the expected number of pages with no mistakes is 1 - e^-a.

I tried using P[D=k|M=m] = P(D=k union M=m)/P(M=m) but don't know where to go from here.
 
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  • #2
P[D=k] = P(D=k union M=m) + P(D=k union M=m+1) + ... I'm not sure how to simplify this expression. Any help would be much appreciated!
 

FAQ: Another probability - random variables, generating functions

What is a random variable?

A random variable is a numerical quantity that takes on different values based on the outcome of a random event. It is used to represent the possible outcomes of a random experiment or situation.

What is a generating function?

A generating function is a mathematical tool used to represent a sequence of numbers or values. It is often used in probability and statistics to find the distribution of a random variable.

How do you calculate the expected value of a random variable?

The expected value of a random variable is calculated by multiplying each possible outcome by its respective probability, and then summing these values. It represents the average value that can be expected from the variable over repeated trials.

What is the difference between discrete and continuous random variables?

A discrete random variable can only take on a finite or countably infinite number of values, while a continuous random variable can take on any value within a certain range. Discrete random variables are typically associated with events that can be counted, while continuous random variables are associated with measurements.

What is the central limit theorem?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will approximate a normal distribution, regardless of the underlying distribution of the individual variables. This theorem is important in statistics, as it allows for the use of normal distribution to analyze data even when the underlying distribution is not known.

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