Consider the following. (a) Suppose a coin that has...

Consider the following. (a) Suppose a coin that has probability $p$ of turning up heads is tossed once. If $X$ is the number of heads, and $Y$ the number of tails show that $X$ and $Y$ are not independent. (b) Let the coin of part (a) be tossed a random number of times $N$, where $N$ is a Poisson random variable with parameter $\alpha$, (see Example 2.7.6 for the definition). Let $X$ and $Y$ be the resulting numbers of heads and tails, respectively. Show that $X$ and $Y$ are independent.

Consider the following.
(a) Suppose a coin that has probability $p$ of turning up heads is tossed once. If $X$ is the number of heads, and $Y$ the number of tails show that $X$ and $Y$ are not independent.
(b) Let the coin of part (a) be tossed a random number of times $N$, where $N$ is a Poisson random variable with parameter $\alpha$, (see Example 2.7.6 for the definition). Let $X$ and $Y$ be the resulting numbers of heads and tails, respectively. Show that $X$ and $Y$ are independent.

Key Concepts

Poisson Splitting Property

The Poisson splitting property is a key concept where, if events from a Poisson process are independently classified into different categories (for example, heads and tails), the counts in each category are independent and themselves follow Poisson distributions with parameters proportional to the original rate. This property underlies the independence of counts in scenarios where the number of trials is random and modeled by a Poisson distribution.

Poisson Distribution

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space when these events occur with a constant mean rate and independently of the time since the last event. It is used for the random counting of events, which plays a critical role when the number of trials itself is random.

Statistical Independence

Statistical independence is a fundamental concept whereby two random variables are independent if the joint probability function can be expressed as the product of their individual marginal probability functions. This means that knowing the outcome of one variable does not provide any information about the outcome of the other.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent and identically distributed Bernoulli trials, each with the same probability of success. In problems with a fixed number of coin tosses, the count of heads (or tails) follows a binomial distribution, highlighting how dependencies arise because the total number of trials is fixed.

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