QUESTION IV Suppose we gather 1000 people on a soccer field and let them stand in a line, each facing the same direction. We give each person a coin. Then, whenever I raise my hand, they will flip their coin (independently) and move according to the following rule: for those whose coin lands Head we will ask them to take a step forwards, while for those with a Tail outcome, we'll ask them to take a step backwards. Say, I raise my hand a large number of time. We might ask what the distribution of their positions will look like. To study the distribution, focus for now on a single individual. Let the random variable Xt represent the movement of the individual at the tth coin flip. We might define Xt as follows: X1 = if Coin lands Head on the tth flip if Coin lands Tail on the tth flip. Now, we can define a new random variable as Y = EXt t=T where T is the number of times I raise my hand. Notice that Y will be the final position of the individual after I raise my hand T times, assuming that the starting position is 0. Now, we have n = 1000 people who'll all flip their coins. So, we can draw a histogram of their positions after different hand-rasiers. Below in Figure 1, I show how the distribu- tion of the 1000 Y values would like after 0, 1, 5, 10, 50, 100, 500, 1000, and 5000 times I raise my hand using computer simulation.Figure 1: Final position of 1, 000 individuals, each moving forward and backward accord- ing to outcome of coin flip whenever I raise my hand # times hand raised = 0 # times hand raised = 1 # times hand raised = 5 0 -1 -5 -3 -1 3 5 # times hand raised = 10 # times hand raised = 50 # times hand raised = 100 -10 10 -30 -20 -10 10 20 -30 -10 0 10 20 30 # times hand raised = 500 # times hand raised = 1000 # times hand raised = 5000 -100 -50 -200 -100 100 200 Before I raise my hand, everyone stays on the same spot (top-left plot in Figure 1). Let this position be 0 (i.e., all individuals start at 0). After one coin-flip, shown in the plot in the middle of the first row of Figure 1, they have either moved one step forward (Y = 1) or one step backward (Y = -1). After five times raising my hand (top-right plot in Figure 1), the smallest possible value is -5, representing individuals whose coin landed Tail five time in a row, and the largest possible value is 5, for those who had five Heads in a row. 1. What is the expected value of Y? (1pt) 2. What is the standard deviation of Y? (1pt) 3. In the second and third plot of the first row of Figure 1, you'll observe that not a single individual's position is exactly equal to zero. Why is this so? (1pt) 4. The distribution looks suspiciously similar to a Normal distribution. Explain why Y cannot be exactly Normally distributed (although we might use the Normal distribution as an approximation). (1pt)5. Re-define X; as if Coin lands Head on the ith flip Xi = if Coin lands Tail on the ith flip. In other words, all individuals who get a Head outcome move one step forward as before; however, for those who get a Tail outcome, they do not move one step backward but stay put. Assume that everything else of this experiment remains the same as the one described above. Again, let Y = > Xt to denote the final position of an individual. Suppose I raise my hand 10, 000 times. What is the distribution of Y? (2pts)