Let A be an event with probability P(A). If E[(IA)
^36217] = Var[2(IA + 36217)]. Find P(A)
To find the probability P(A) using the given information, we need to apply the properties of expected value and variance.
The expected value (E) and variance (Var) are statistical measures that describe the behavior of a random variable.
Given:
E[(IA)^36217] = Var[2(IA + 36217)]
Let's break down the equation step by step:
Step 1: Simplify the expressions inside the expected value and variance.
E[(IA)^36217] = E[I^36217 * A^36217]
Var[2(IA + 36217)] = Var[2IA + 2 * 36217]
Step 2: Apply the properties of expected value and variance.
E[cX] = c * E[X] --> Property 1 (where c is a constant)
Var[cX] = c^2 * Var[X] --> Property 2 (where c is a constant)
Using Property 1, we can rewrite the equation as:
E[I^36217] * E[A^36217] = Var[2IA] + Var[2 * 36217]
Step 3: Since I is an indicator variable, it can only take the values 0 or 1. Therefore, I^36217 is always equal to I.
E[I] * E[A^36217] = Var[2IA] + Var[2 * 36217]
Step 4: Apply Property 1 to the left side of the equation.
E[A^36217] = Var[2IA] + Var[2 * 36217] / E[I]
Step 5: Since I is an indicator variable, E[I] is equal to the probability of event A, which is P(A).
E[A^36217] = Var[2IA] + Var[2 * 36217] / P(A)
Step 6: Simplify Var[X] = E[X^2] - E[X]^2 (variance formula).
E[A^36217] = Var[2IA] + Var[2 * 36217] / P(A)
E[A^36217] = E[(2IA)^2] - E[2IA]^2 + Var[2 * 36217] / P(A)
Step 7: Simplify the expressions inside the expected value.
E[(2IA)^2] = E[4I^2 * A^2]
= 4 * E[I^2] * E[A^2] --> Property 1
E[2IA] = 2 * E[I] * E[A] --> Property 1
Step 8: Plug in the simplified expressions.
E[A^36217] = 4 * E[I^2] * E[A^2] - (2 * E[I] * E[A])^2 + Var[2 * 36217] / P(A)
Step 9: Since I is an indicator variable, I^2 is always equal to I itself.
E[A^36217] = 4 * E[I] * E[A^2] - (2 * E[I] * E[A])^2 + Var[2 * 36217] / P(A)
Step 10: Since I is an indicator variable, E[I] is equal to the probability of event A, which is P(A).
E[A^36217] = 4 * P(A) * E[A^2] - (2 * P(A) * E[A])^2 + Var[2 * 36217] / P(A)
Step 11: Simplify Var[2 * 36217] = Var[72434].
Since 72434 is a constant, Var[cX] = c^2 * Var[X], where c is a constant.
Var[72434] = Var[1] * 72434^2 = 72434^2 * Var[1] = 72434^2 * 0 = 0
E[A^36217] = 4 * P(A) * E[A^2] - (2 * P(A) * E[A])^2 + 0 / P(A)
Step 12: Since the variance of a constant is 0, we can ignore the last term.
E[A^36217] = 4 * P(A) * E[A^2] - (2 * P(A) * E[A])^2
Step 13: Notice that E[A^36217] is equal to the second moment of A raised to the power of 36217, denoted as μ_2(A^36217).
μ_2(A^36217) = 4 * P(A) * E[A^2] - (2 * P(A) * E[A])^2
Step 14: Finally, to find P(A), divide both sides by μ_2(A^36217).
P(A) = μ_2(A^36217) / [4 * E[A^2] - (2 * E[A])^2]
Therefore, P(A) can be found by computing the second moment of A raised to the power of 36217, denoted as μ_2(A^36217), and then plugging it into the formula above along with the values of E[A^2] and E[A].