let X be a geometric random variable, and assume that Var(X) = 5
a) what is the conditional variance Var(X-4 | X>4)?
b) what is the conditional variance Var(X-8 | X>4)?
a) V(x-4|x>4)=V(x-4)
=V(x)
=5
b)V(x-8|x>4)=V(x-8)
=V(x)
=5
a) It seems like X is feeling a bit cramped and wants to get some space. So, let's calculate the conditional variance Var(X-4 | X>4) to give X a little breathing room.
Now, since X is a geometric random variable, let's break it down. The variance of X, Var(X), is given as 5. But what does it mean for X to be geometric? Well, it means that X represents the number of trials needed to achieve the first success in a series of independent trials with a specific probability of success.
So, to calculate the conditional variance Var(X-4 | X>4), we need to take into account that X is greater than 4. In other words, we are only considering the trials after the 4th trial.
Now, since X is a geometric random variable, each trial is independent of the others. Therefore, the variance of X-4 (i.e., the number of trials after the 4th trial) will be the same as the variance of X.
So, the conditional variance Var(X-4 | X>4) is equal to Var(X), which is 5. X finally has some space to stretch its legs!
b) Now X wants to take a bigger leap! Let's calculate the conditional variance Var(X-8 | X>4) to see how far X can go.
Since X>4, we know that X is already past the 4th trial. But now, X wants to go even further and reach the 8th trial.
The difference X-8 represents the number of trials after the 8th trial. Again, since X is a geometric random variable and each trial is independent, the variance of X-8 will be the same as the variance of X.
So, the conditional variance Var(X-8 | X>4) is also equal to Var(X), which is still 5. X is feeling adventurous and ready to go the extra mile!
Remember, X is a geometric random variable, but don't worry, it's not all Greek to me.
To solve these problems, we will use conditional variance formula:
Var(Y | A) = Var(Y) / P(A)
where Y is a random variable, A is the event, and P(A) is the probability of event A.
Let's solve each part step-by-step:
a) To find Var(X-4 | X>4), we need to find the conditional variance of X-4 given that X>4.
Step 1: Find P(X>4)
To find P(X>4), we can use the formula for the probability of a geometric random variable:
P(X=k) = (1-p)^(k-1) * p
In this case, we need to find P(X>4), which is the sum of all the probabilities when X is greater than 4.
P(X>4) = P(X=5) + P(X=6) + ...
We know that Var(X) = 5, so p = 1/sqrt(5)
P(X>4) = (1-1/sqrt(5))^4 + (1-1/sqrt(5))^5 + ...
This is an infinite geometric series. To find the sum, we can use the formula:
P(X>4) = (1-1/sqrt(5))^4 / (1 - (1-1/sqrt(5)))
Step 2: Find Var(X-4)
To find Var(X-4), we need to find E[(X-4)²] - E[X-4]².
E[X-4] is the expected value of X-4, which can be found using the expected value formula:
E[X-4] = E[X] - 4
E[(X-4)²] can be found using the variance formula:
E[(X-4)²] = Var(X) + E[X]² - 2E[X] * 4 + 4²
Substituting the given value of Var(X) = 5, we can calculate E[(X-4)²] and E[X-4]².
Step 3: Calculate Var(X-4 | X>4)
Finally, we can calculate Var(X-4 | X>4) using the conditional variance formula:
Var(X-4 | X>4) = Var(X-4) / P(X>4)
Substitute the calculated values of Var(X-4) and P(X>4) to obtain the final answer.
b) To find Var(X-8 | X>4), we follow the same steps as in part a.
Step 1: Calculate P(X>4)
Step 2: Find Var(X-8)
Step 3: Calculate Var(X-8 | X>4) using the conditional variance formula.
Substitute the calculated values of Var(X-8) and P(X>4) to obtain the final answer.
To find the conditional variance Var(X-4 | X>4) and Var(X-8 | X>4), we need to first understand the concept of conditional variance and how it is calculated.
In general, conditional variance (Var(Y|X)) measures the variability or spread of a random variable Y, given that another random variable X has a specific value or falls within a certain range. It is defined as the expected value of the squared difference between Y and its conditional mean, given X.
a) Let's start with the conditional variance Var(X-4 | X>4):
To calculate this, we need to find the conditional mean of (X-4 | X>4) first.
1. Find the conditional mean E(X-4 | X>4):
The conditional mean E(X-4 | X>4) is the expected value of (X-4) given that X>4.
Since X is a geometric random variable, the conditional mean formula for a geometric random variable is given by:
E(X | X>4) = (p / (1 - (1-p)^4)), where p is the probability of success in each trial.
In this case, we are looking for the conditional mean of (X-4), so the formula becomes:
E(X-4 | X>4) = [(p / (1 - (1-p)^4))] - 4.
2. Find the conditional variance Var(X-4 | X>4):
The conditional variance Var(X-4 | X>4) is the expected value of the squared difference between (X-4) and its conditional mean.
Var(X-4 | X>4) = E[((X-4) - E(X-4 | X>4))^2]
Substituting the conditional mean we found earlier:
Var(X-4 | X>4) = E[((X-4) - [(p / (1 - (1-p)^4))] + 4)^2]
b) Now let's move on to the conditional variance Var(X-8 | X>4):
To calculate this, we follow a similar process as above, but this time we use (X-8) instead of (X-4).
1. Find the conditional mean E(X-8 | X>4):
E(X-8 | X>4) = [(p / (1 - (1-p)^4))] - 8.
2. Find the conditional variance Var(X-8 | X>4):
Var(X-8 | X>4) = E[((X-8) - E(X-8 | X>4))^2]
Substituting the conditional mean we found earlier:
Var(X-8 | X>4) = E[((X-8) - [(p / (1 - (1-p)^4))] + 8)^2]
Please note that to find the exact values of the conditional variance, you would need to know the specific value of the probability of success (p) in each trial.