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The range of a random variable X is {1, ...

The range of a random variable X is {1, 2, 3, ……..} and `P(X=k)=(C^(k))/(lfloork)` where k = 1, 2, 3,…. Find the value of C

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To find the value of \( C \) in the given probability distribution function \( P(X = k) = \frac{C^k}{k!} \), we will follow these steps: ### Step 1: Set up the equation for the sum of probabilities The sum of the probabilities for all possible values of \( k \) must equal 1. Therefore, we write: \[ \sum_{k=1}^{\infty} P(X = k) = 1 \] Substituting the expression for \( P(X = k) \): \[ \sum_{k=1}^{\infty} \frac{C^k}{k!} = 1 \] ### Step 2: Recognize the series The series \( \sum_{k=0}^{\infty} \frac{C^k}{k!} \) is known to be the Taylor series expansion for \( e^C \). However, since our series starts from \( k=1 \), we can express it as: \[ \sum_{k=1}^{\infty} \frac{C^k}{k!} = e^C - 1 \] ### Step 3: Set the equation equal to 1 From the previous step, we have: \[ e^C - 1 = 1 \] ### Step 4: Solve for \( e^C \) Adding 1 to both sides gives: \[ e^C = 2 \] ### Step 5: Take the natural logarithm To solve for \( C \), we take the natural logarithm of both sides: \[ C = \ln(2) \] ### Final Result Thus, the value of \( C \) is: \[ C = \ln(2) \] ---
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