Question

1. Multi-millionaire lottery Suppose you bought a lottery ticket at $100 today, January 5, 2018. You will find out the outcome on January 5, 2019. Let X be the possible dollar amount of the payoff you may receive on January 5, 2019 from the lottery. Assume the prevailing interest rate over the next 10 years is flat at 5% (per year). The likelihood of the possible outcome is assumed as follows. 4 Prob(X) 11/369/36 7/36 5/36 3/36 1/36 (1) What is your expected payoff on January 5, 2019? (2) What are the variance /standard deviation of your expectation in (1)? (3) What is vour exnected gain or loss on January 5. 2019?

Answer 1

Que 1

Expected payoff is the sum of probability adjusted payoff across all the possible outcomes = X_e =

$\sum_{i=1}^{6} P_{i}\times X_{i}$ where P_i is the probability of payoff Xi

= 1 x 11/36 + 2 x 9/36 + 3 x 7/36 + 4 x 5/36 + 5 x 3/36 + 6 x 1 / 36 = $ 91 / 36 mn = $ 2.53 mn

Que 2

Variance of discrete distribution, $\sigma$²

$= \sum_{i=1}^{6} P_{i}\times (X_{i} - X_{e})^{2}$ where X_e = expected payoff as calculated in Que 1 above

= (1 - 2.53)² x 11/36 + (2 - 2.53)² x 9/36 + (3 - 2.53)² x 7/36 + (4 - 2.53)² x 5/36 + (5 - 2.53)² x 3/36 + (6 - 2.53)² x 1 / 36 = 1.971451

Standard deviation, $= \sigma$ = $= \sqrt{\sigma^{2}} = \sqrt{1.971451}$ = $ 1.40 mn

Que 3

Expected gain / loss = Expected payoff - cost of the lottery ticket = $ 2,527,778 - 100 = $ 2,527,678

Que 4

Future compounded expected payoff = Future Value of expected payoff X_e = X_e x (1 + R)^N where R = interest rate = 5% per annum and N = period = 5 years

Hence, Future compounded expected payoff = 2.53 x (1 + 5%)⁵ = $ 3.23 mn

Que 5

Yes, there is a correlation between the payoff amount X and the probability associated with it. The probability of the payoff declines as payoff amount increase.

Que 6

As X increases, probability decreases. So, if probability is dependent variable and payoff, X is independent variable then, the depended variable decreases in value as the value of independent variable increase. So, the sign of the regression coefficient must be negative.

Que 7

Regression coefficient is the coefficient of independent variable in the equation of regression. If Y = aX + b is the linear regression equation connecting X and Y,  then Y is the dependent variable and X is the independent variable. Regression coefficient will then be "a", the coefficient of X in the equation of regression. Thus regression coefficient measures the change in the value of dependent variable for a unit change in the value of independent variable. Thus it is also an indicator of sensitivity of dependent variable with respect to independent variable.

Que 8

P value test tells us whether the regression coefficients are significant or not.

The P value in turn is dependent upon t statistics and degrees of freedom.

If P value is less than the common alpha level of 0.05, then the coefficient is statistically significant otherwise it's not.