To find the value of [tex]\( k \)[/tex] and calculate the mean of the given probability distribution, follow these steps:
### Step 1: Determine [tex]\( k \)[/tex]
The sum of all probabilities in a probability distribution must equal 1. Therefore, we'll set up the equation using the probabilities given:
[tex]\[ 0.1 + k + 0.2 + 2k + 0.3 + k = 1 \][/tex]
Combine like terms:
[tex]\[ 0.1 + 0.2 + 0.3 + k + 2k + k = 1 \][/tex]
[tex]\[ 0.6 + 4k = 1 \][/tex]
Isolate [tex]\( k \)[/tex] by subtracting 0.6 from both sides:
[tex]\[ 4k = 1 - 0.6 \][/tex]
[tex]\[ 4k = 0.4 \][/tex]
Now, divide by 4:
[tex]\[ k = \frac{0.4}{4} \][/tex]
[tex]\[ k = 0.1 \][/tex]
### Step 2: Verify the Probabilities
Using [tex]\( k = 0.1 \)[/tex], substitute back into the original probabilities:
- [tex]\( p(-2) = 0.1 \)[/tex]
- [tex]\( p(-1) = 0.1 \)[/tex]
- [tex]\( p(0) = 0.2 \)[/tex]
- [tex]\( p(1) = 2k = 2 \times 0.1 = 0.2 \)[/tex]
- [tex]\( p(2) = 0.3 \)[/tex]
- [tex]\( p(3) = 0.1 \)[/tex]
Sum these probabilities to ensure they equal 1:
[tex]\[ 0.1 + 0.1 + 0.2 + 0.2 + 0.3 + 0.1 = 1 \][/tex]
The total is indeed 1, so [tex]\( k = 0.1 \)[/tex] is confirmed.
### Step 3: Calculate the Mean (Expected Value)
The mean [tex]\( \mu \)[/tex] of a discrete random variable is given by:
[tex]\[ \mu = \sum (x \cdot p(x)) \][/tex]
Substitute the values of [tex]\( x \)[/tex] and corresponding probabilities [tex]\( p(x) \)[/tex]:
[tex]\[ \mu = (-2 \cdot 0.1) + (-1 \cdot 0.1) + (0 \cdot 0.2) + (1 \cdot 0.2) + (2 \cdot 0.3) + (3 \cdot 0.1) \][/tex]
Perform the multiplications:
[tex]\[ \mu = (-0.2) + (-0.1) + (0) + (0.2) + (0.6) + (0.3) \][/tex]
Add these values:
[tex]\[ \mu = -0.2 - 0.1 + 0 + 0.2 + 0.6 + 0.3 \][/tex]
Combine the terms:
[tex]\[ \mu = 0.8 \][/tex]
### Conclusion
The value of [tex]\( k \)[/tex] is [tex]\( 0.1 \)[/tex], and the mean of the distribution is [tex]\( 0.8 \)[/tex].
