Lottery-type stochastics question

[sunrah]

Homework Statement

n balls, numbered from 1 to n, are drawn randomly from an urn one after the other (whilst returning each ball before drawing the next). The random variable T_k represents the number of draws until k different balls have been drawn, where k = 1,..., n.

calculate:
a) P[T₁ = m₁ ^ T₂ - T₁ = m₂ ^ . . . ^ T_n - T_n-1 = m_n]

for m_i in N.

b) show that random variables T₁, T₂ - T₁, . . . , T_n - T_n-1 are independent

Homework Equations

The Attempt at a Solution

a)
P[T₁ = m₁ ^ T₂ - T₁ = m₂ ^ . . . ^ T_n - T_n-1 = m_n]

= P[m₁ ^ m₂ ^ . . . ^ m_n] = P(m₁)P(m₂)...P(m_n) = $\prod^{n}_{i=1} P(m_{i})$

ok so i think I have to work out the product from i = 1,..,n of the probabilities P(m_i) but how I don't even know what the m_i are.

b) I guess this means linearly independent, I don't know?

 

[Ray Vickson]

Homework Statement

n balls, numbered from 1 to n, are drawn randomly from an urn one after the other (whilst returning each ball before drawing the next). The random variable T_k represents the number of draws until k different balls have been drawn, where k = 1,..., n.

calculate:
a) P[T₁ = m₁ ^ T₂ - T₁ = m₂ ^ . . . ^ T_n - T_n-1 = m_n]

for m_i in N.

b) show that random variables T₁, T₂ - T₁, . . . , T_n - T_n-1 are independent

Homework Equations

The Attempt at a Solution

a)
P[T₁ = m₁ ^ T₂ - T₁ = m₂ ^ . . . ^ T_n - T_n-1 = m_n]

= P[m₁ ^ m₂ ^ . . . ^ m_n] = P(m₁)P(m₂)...P(m_n) = $\prod^{n}_{i=1} P(m_{i})$

ok so i think I have to work out the product from i = 1,..,n of the probabilities P(m_i) but how I don't even know what the m_i are.

b) I guess this means linearly independent, I don't know?

Presumably, $T_1=1,$ because the definition you gave would be "$T_1 =$ the number of draws until 1 ball is obtained". Is that correct? (Of course, a deterministic quantity is a random variable with a degenerate probability distribution.) So, translating: $T_2-1$ is the number of draws until we get a ball different from the first one, $T_3-T_2$ is the number of draws until we get a ball different from the first two, etc. The problem is asking for the joint probability distribution of these numbers, and is then asking you to prove that these numbers are independent random variables. Linear independence has absolutely nothing to do with the problem.

To get started, I suggest you look first at the distribution of $T_2-1$ (it is a distribution you should have seen already!), then look at the distribution of $T_3 - T_2,$ given a value $\{T_1-1 = m_2\},$; that is, you need to get
$$P\{T_3 - T_2 = m_3 | T_2 - 1 = m_2\}$$ This should be a very familiar distribution. Note: here, $m_2, \, m_3 = 1, 2, 3, \ldots$ are not specified; you need to work out what happens for any possible values of $m_2, \, m_3.$

After you have done these two steps you should see the pattern that emerges.

RGV