91Ó°ÊÓ

In probability theory, the concept of independent random variables is crucial. Independent random variables are those whose occurrences do not affect each other. For instance, when tossing a biased coin multiple times, the result of one toss (e.g., head or tail) does not alter the outcome probabilities of subsequent tosses.

In the provided exercise, we consider the number of heads (H) and tails (T) from coin tosses as independent random variables. This arises because if the number of coin tosses N follows a Poisson distribution, then H and T, derived from N, are also Poisson distributed. However, they each have their own parameters:

• The number of heads H has a parameter \( \lambda p \).
• The number of tails T has a parameter \( \lambda (1-p) \).

The independence between H and T stems from the structure of Poisson distribution. Since H and T are derived independently from the same pool of trials, knowing the number of heads does not influence the probability of the number of tails, affirming their statistical independence.

Calculating variance is essential when dealing with random variables as it measures how much the values disperse from the expected value. For Poisson random variables, variance has a unique simplicity: it is equal to the parameter of the distribution.

In this exercise, the random variables H (heads) and T (tails) are each distributed as Poisson with parameters \(\lambda p\) and \( \lambda(1-p) \), respectively. Therefore:

• The variance of H is \( Var(H) = \lambda p \)
• The variance of T is \( Var(T) = \lambda(1-p) \)

Furthermore, when determining the variance of the difference H - T, the principle used is that the variance of a difference between two independent variables is the sum of their variances.

• Thus, \( Var(H - T) = Var(H) + Var(T) = \lambda p + \lambda (1-p) = \lambda \)

This calculation reveals how variance helps us understand the variability or spread of the difference of heads and tails despite being separate in occurrence.

Expected value, often considered as the mean, provides a measure of the central tendency or average outcome one can expect from a random variable. For a Poisson random variable, the expected value is directly its parameter.

In this exercise scenario, the expected number of heads (H) and tails (T) from N biased coin tosses are:

- \( E[H] = \lambda p \), signifying the average number of heads.- \( E[T] = \lambda (1-p) \), indicating the average number of tails.To find the expected value for the difference H - T (i.e., heads minus tails), we use the fact that expected values add linearly even for differences, which gives us:

• \( E[H - T] = E[H] - E[T] = \lambda p - \lambda (1-p) = \lambda(2p-1) \)

This expected value helps describe the average difference between the number of heads and tails for N tosses, and provides insight into which outcome (heads or tails) is dominant based on the initial bias (probability p) of the coin.