Let Y be the difference between the number of heads and the number of tails in the 3
tosses of a fair coin.
(a) Plot the cdf of the random variable Y.
(b) Express P[|Y|<y] in terms of the cdf of Y
To answer the question, we need to understand what a fair coin is and how to calculate the difference between the number of heads and tails in three coin tosses. Let's break down the steps:
Step 1: Understanding a fair coin
A fair coin is a coin where the probability of landing on heads and tails is equal, both 0.5 or 50%. This means that for every coin toss, there is an equal chance of getting heads or tails.
Step 2: Calculating the difference between the number of heads and tails
In three coin tosses, there are various possible outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. We need to calculate the difference between the number of heads and tails in each outcome.
(a) Plotting the cumulative distribution function (CDF) of random variable Y
To plot the CDF of Y, we need to calculate the probability of obtaining each possible value of Y up to a given point.
The possible values of Y are -3, -2, -1, 0, 1, 2, 3, since the maximum difference we can get is 3 (3 heads - 0 tails) and the minimum is -3 (0 heads - 3 tails).
For example, to find the probability of Y = 1, we need to count the number of outcomes where the difference is 1: HHT, HTH, THH. Since each toss is independent and has a 0.5 probability, the probability of getting 1 head and 2 tails is (0.5 * 0.5 * 0.5) * 3 = 0.375.
Similarly, we calculate the probabilities for each possible value of Y:
P(Y = -3) = P(0 heads - 3 tails) = (0.5 * 0.5 * 0.5) = 0.125
P(Y = -2) = P(1 head - 2 tails) = (0.5 * 0.5 * 0.5) * 3 = 0.375
P(Y = -1) = P(2 heads - 1 tail) = (0.5 * 0.5 * 0.5) * 3 = 0.375
P(Y = 0) = P(3 heads - 3 tails) = (0.5 * 0.5 * 0.5) = 0.125
P(Y = 1) = P(3 heads - 2 tails) = (0.5 * 0.5 * 0.5) * 3 = 0.375
P(Y = 2) = P(3 heads - 1 tail) = (0.5 * 0.5 * 0.5) * 3 = 0.375
P(Y = 3) = P(3 heads - 0 tails) = (0.5 * 0.5 * 0.5) = 0.125
Now, we can plot the CDF of Y by summing up the probabilities for each value of Y up to a given point.
(b) Expressing P[|Y|<y] in terms of the CDF of Y
P[|Y|<y] represents the probability that the absolute value of Y is less than y. This means we need to find the probability that Y takes on the values -2, -1, 0, 1, or 2, depending on the value of y.
To express P[|Y|<y] in terms of the CDF of Y, we need to consider the cumulative probabilities up to y for both positive and negative values of Y.
For example, if y = 2, P[|Y|<2] means we need to consider Y = -1, 0, 1. We can calculate it as:
P[|Y|<2] = P(Y = -1) + P(Y = 0) + P(Y = 1)
By using the cumulative probabilities calculated for the CDF, we can express P[|Y|<y] in terms of the CDF of Y.
Note: The calculations provided above assume a fair coin, where the probability of getting heads or tails in each toss is 0.5.