The value of C for which `P(X=k)=Ck^2` can serve as the probability function of a random varibale X that takes value `0,1,2,3,4,`is

Options

A. `(1)/(30)`
B. `(1)/(10)`
C. `(1)/(3)`
D. `(1)/(15)`

Answer

To find the value of \( C \) for which \( P(X=k) = Ck^2 \) can serve as the probability function of a random variable \( X \) that takes values \( 0, 1, 2, 3, 4 \), we need to follow these steps:

### Step 1: Understand the Probability Function
The probability function is given as \( P(X=k) = Ck^2 \). The values of \( k \) can be \( 0, 1, 2, 3, 4 \).

### Step 2: Set Up the Equation
The sum of the probabilities for all possible values of \( k \) must equal \( 1 \):
\[ \sum_{k=0}^{4} P(X=k) = 1 \]
Substituting the probability function:
\[ \sum_{k=0}^{4} Ck^2 = 1 \]

### Step 3: Calculate the Individual Probabilities
Now we calculate \( P(X=k) \) for each value of \( k \):
- For \( k=0 \): \( P(X=0) = C(0^2) = 0 \)
- For \( k=1 \): \( P(X=1) = C(1^2) = C \)
- For \( k=2 \): \( P(X=2) = C(2^2) = 4C \)
- For \( k=3 \): \( P(X=3) = C(3^2) = 9C \)
- For \( k=4 \): \( P(X=4) = C(4^2) = 16C \)

### Step 4: Sum the Probabilities
Now we sum these probabilities:
\[ P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) = 0 + C + 4C + 9C + 16C \]
This simplifies to:
\[ 0 + C + 4C + 9C + 16C = 30C \]

### Step 5: Set the Equation Equal to 1
We set the sum equal to \( 1 \):
\[ 30C = 1 \]

### Step 6: Solve for C
Now, we solve for \( C \):
\[ C = \frac{1}{30} \]

### Conclusion
Thus, the value of \( C \) for which \( P(X=k) = Ck^2 \) serves as a valid probability function is:
\[ \boxed{\frac{1}{30}} \]