Suppose a person buys 100 shares of stock at in a company at the cost price of $10 per share. He plans to sell at the end of the next month at whatever is the price at that time. The projected price of the stock at the end of next month and the corresponding probability of reaching that price are given below:
Projected Price Probability
$8.00 0.10
$9.00 0.20
$10.00 0.25
$11.00 0.30
$12.00 0.15
Suppose X denotes the profit he will make from the sale of the 100 shares.
• What is the expected value of X?
• What is the variance of X?
expect value x
a. 8(0.1) + 9(.2) + 10(.25) + 11(.30) + 12 (.15) = 10.2
b. 8^2(0.1) + 9^2(.2) + 10^2(.25) + 11^2(.30) + 12^2(.15) = 180.5
Are you sure that is the variance. I got .14
To calculate the expected value and variance of X, we need to multiply each possible outcome by its corresponding probability and then sum up these values.
Step 1: Calculate the expected value of X
To find the expected value of X, we multiply each possible outcome by its corresponding probability and sum them up.
Expected value (E) = (Value1 × Probability1) + (Value2 × Probability2) + ... + (ValueN × ProbabilityN)
In this case, the possible outcomes are the projected prices, and their corresponding probabilities are given. We will multiply each projected price by its probability and sum them up.
Expected value (E) = ($8.00 × 0.10) + ($9.00 × 0.20) + ($10.00 × 0.25) + ($11.00 × 0.30) + ($12.00 × 0.15)
E = $0.80 + $1.80 + $2.50 + $3.30 + $1.80
E = $9.20
Therefore, the expected value of X is $9.20.
Step 2: Calculate the variance of X
To find the variance of X, we need to calculate the squared difference between each possible outcome and the expected value, multiply it by its corresponding probability, and sum them up.
Variance (Var) = [(Value1 - E)^2 × Probability1] + [(Value2 - E)^2 × Probability2] + ... + [(ValueN - E)^2 × ProbabilityN]
Using the projected prices and their probabilities:
Var = [(8.00 - 9.20)^2 × 0.10] + [(9.00 - 9.20)^2 × 0.20] + [(10.00 - 9.20)^2 × 0.25] + [(11.00 - 9.20)^2 × 0.30] + [(12.00 - 9.20)^2 × 0.15]
Var = [(-1.20)^2 × 0.10] + [(-0.20)^2 × 0.20] + (0.80^2 × 0.25) + (1.80^2 × 0.30) + (2.80^2 × 0.15)
Var = 1.44 × 0.10 + 0.04 × 0.20 + 0.64 × 0.25 + 3.24 × 0.30 + 7.84 × 0.15
Var = 0.144 + 0.008 + 0.16 + 0.972 + 1.176
Var = 2.46
Therefore, the variance of X is 2.46.
In summary:
- The expected value of X is $9.20.
- The variance of X is 2.46.