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 find the expected value of a random variable, we need to multiply each possible outcome by its corresponding probability and then add them up.
In this case, we have a fair 10-sided die, so each side has a probability of 1/10 of being rolled.
Let's consider the possible outcomes for X, the number of tosses until getting a 4:
- If we get a 4 on the first toss, X is 1.
- If we get a number between 1 and 3 on the first toss, we need to consider the expected number of tosses until getting a 4, which is X+1.
- If we get a number between 5 and 10 on the first toss, we need to consider the expected number of tosses until getting a 4, which is X+1.
So, the expected value E(X) can be calculated as follows:
E(X) = (1/10 * 1) + (3/10 * (X+1)) + (6/10 * (X+1))
Simplifying this equation, we can rewrite it as:
E(X) = (1/10) + (3/10)(X+1) + (6/10)(X+1)
Multiplying each term by their probabilities, we have:
E(X) = 1/10 + (3/10)(X+1) + (6/10)(X+1)
= 1/10 + (3/10)X + 3/10 + (6/10)X + 6/10
= (8/10) + (9/10)X
Therefore, the expected value of X, denoted as E(X), is (8/10) + (9/10)X.