Problem 0
You are playing a game at a carnival, the game involves rolling a 3 sided dice where: Rolling a
- 1 has probability of 2 / 10
- 2 has probability of 5 / 10
- 4 has probability of 3 / 10
If you land on the dice face 2, you get to flip a fair coin otherwise you will flip an unfair coin such that P(Tails)=.30 Let X be a random variable that represents the number of dots you see on a face of the dice, and Y be an indicator variable on whether the coin shows a head when flipped.
Find var(X+4Y) analytically and confirm via simulation
The simulation found similar variance
Problem 1:
Consider the bus ridership example. Intuitively, L1 and L2 are not independent, show that var(L1-L2)does not equal Var(L1)+ var(L2). Determine the difference. (Find the three variances analytically, and confirm via simulation.)
The final variance should be about 0.55, but I could not work out a few problems in my arithmetic.
Problem 2:
The game is to toss a coin until we get r consecutive heads or reach a total of s tosses, whichever comes first. Let X denote the number of tosses we make. We win $X. Find the minimum fee that should be charged for this game if r = 4 and s = 7. Confirm via simulation.
This means the minimum fee we should charge to play the game is slightly under this, or $6.71.
Problem 3:
Let X and Y be indicator random variables such that P(X = 1), P(Y =1) and P(X = Y = 1) are p, q and r, respectively. Find Var(3X-2Y), as a function of p, q and r. NO SIMULATION
Problem 4:
Let X be a random variable that denotes the sum of the values on a roll of 2 dice(8 sided dice with equal prob of getting any face (1-8)).
- What values does the random variable take?
- Find the pmf.
- What is the expected value of X? Confirm via simulation.
- What is the variance ?
The simulation found a similar variance