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(Lamperti<ref group="Notes" >Private communication.</ref>) An urn contains exactly 5000 balls, of which an unknown number <math>X</math> are white and the rest red, where <math>X</math> is a random variable with a probability distribution on the integers 0, 1, 2, ..., 5000. | |||
<ul style="list-style-type:lower-alpha"><li> Suppose we know that <math>E(X) = \mu</math>. Show that this is enough to allow us to | |||
urn contains exactly 5000 balls, of which an unknown number <math>X</math> are white and the rest red, where <math>X</math> | |||
is a random variable with a probability distribution on the integers 0, 1, 2, | |||
<ul><li> Suppose we know that <math>E(X) = \mu</math>. Show that this is enough to allow us to | |||
calculate the probability that a ball drawn at random from the urn will be white. | calculate the probability that a ball drawn at random from the urn will be white. | ||
What is this probability? | What is this probability? | ||
Latest revision as of 21:21, 14 June 2024
(Lamperti[Notes 1]) An urn contains exactly 5000 balls, of which an unknown number [math]X[/math] are white and the rest red, where [math]X[/math] is a random variable with a probability distribution on the integers 0, 1, 2, ..., 5000.
- Suppose we know that [math]E(X) = \mu[/math]. Show that this is enough to allow us to calculate the probability that a ball drawn at random from the urn will be white. What is this probability?
- We draw a ball from the urn, examine its color, replace it, and then draw
another. Under what conditions, if any, are the results of the two drawings
independent; that is, does
[[math]] P({{\rm white},{\rm white}}) = P({{\rm white}})^2\ ? [[/math]]
- Suppose the variance of [math]X[/math] is [math]\sigma^2[/math]. What is the probability of drawing two white balls in part (b)?
Notes