Answer:
E[W] = $25 (assuming the currency is in dollars)
Var(W) = 1041.67
Step-by-step explanation:
Probability of winning first starts with the coin toss.
For a win, the coin needs to land on heads.
Probability of that = 1/2 = 0.5
Then probability of winning any amount = 1/100 = 0.01
Total probability of winning any amount = 0.5 × 0.01 = 0.005
But expected value is given by
E(X) = Σ xᵢpᵢ
where xᵢ is each amount that could be won
pᵢ is the probability of each amount to be won and it is the same for all the possible winnings = 0.005
So,
E(W) = Σ 0.005 xᵢ
Summing from 0 to 100 (0 indicating getting a tail from the coin toss). This could be done with dome faster with an integral sign
E(W) = ∫ 0.005 x dx
Integrating from 0 to 100
E(W) = [0.005 x²/2]¹⁰⁰₀
E(W) = [0.0025 x²]¹⁰⁰₀ = 0.0025(100² - 0²) = 0.0025 × 10000 = $25
Variance is given by
Variance = Var(X) = Σxᵢ²pᵢ − μ²
μ = expected value
We calculate the expression, Σxᵢ²pᵢ which is another sum from 0 to 100
Σxᵢ²pᵢ = Σ 0.005xᵢ²
Σ 0.005 xᵢ² = ∫ 0.005 x² dx
Integrating from 0 to 100
∫ 0.005 x² dx = [0.005 x³/3]¹⁰⁰₀ = [0.1667x³]¹⁰⁰₀ = 0.1667(100³ - 0³) = 1666.67
Var(W) = 1666.67 - 25² = 1666.67 - 625 = 1041.67.
