Compute E(X) for the following random variable X :
X=Number of tosses until getting 4 (including the last toss) by tossing a fair 10-sided die.
E(X)=...............
To compute E(X) for the given random variable X, which represents the number of tosses until getting a 4 by tossing a fair 10-sided die, we need to find the expected value.
To understand how to compute E(X) in this case, we will break down the problem into steps:
Step 1: Determine the probability of getting a 4 on a single toss.
Since we are tossing a fair 10-sided die, the probability of obtaining a 4 on a single toss is 1/10 or 0.1.
Step 2: Determine the probabilities of not getting a 4 on each toss.
The probability of not getting a 4 on a single toss is 9/10 or 0.9. This is because there are 9 other outcomes (1, 2, 3, 5, 6, 7, 8, 9, and 10) out of the 10 possible outcomes that are not a 4.
Step 3: Analyze the pattern of probabilities.
To find the expected value, we need to consider the pattern of probabilities for each toss until we get a 4. Let's look at a few examples:
- On the first toss, the probability of getting a 4 is 0.1.
- If we do not get a 4 on the first toss, we have to try again. The probability of not getting a 4 on the first toss and then getting a 4 on the second toss is (0.9) * (0.1) = 0.09.
- If we do not get a 4 on the first two tosses, we have to try again. The probability of not getting a 4 on the first two tosses and then getting a 4 on the third toss is (0.9) * (0.9) * (0.1) = 0.081.
- We continue this pattern until we eventually get a 4.
Step 4: Calculate the expected value.
To calculate E(X), we need to multiply the number of tosses in each scenario by their respective probabilities and sum them up. In this case, we will continue until we reach the expected value.
E(X) = (1 * 0.1) + (2 * 0.09) + (3 * 0.081) + (4 * 0.073) + ...
This is an infinite series called the geometric series, where each term is multiplied by a common ratio of 0.9. The formula to calculate the sum of a geometric series is:
Sum = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio.
In this case, the first term (a) is 1 and the common ratio (r) is 0.9. Plugging these values into the formula:
E(X) = 1 / (1 - 0.9)
E(X) = 1 / 0.1
E(X) = 10
Therefore, the expected value of X, E(X), is 10.