x 1 2 3 4 5
P(X=x) 0.1 0.2 a b 0.2
Given that E(X)=3.1, calculate the value of a and b
To calculate the values of a and b, we will use the expected value (E) formula. The expected value of a random variable X is calculated by summing the product of each possible value of X and its corresponding probability.
The formula for the expected value is:
E(X) = x₁ * P(X=x₁) + x₂ * P(X=x₂) + ... + xn * P(X=xn)
Given that E(X) = 3.1, we can plug in the values from the given table:
3.1 = 1 * 0.1 + 2 * 0.2 + 3 * a + 4 * b + 5 * 0.2
Simplifying this equation:
3.1 = 0.1 + 0.4 + 3a + 4b + 1
Combine like terms:
3.1 = 3a + 4b + 1.5
Rearranging the equation:
3a + 4b = 3.1 - 1.5
3a + 4b = 1.6
Now, we have a linear equation with two variables. To solve for a and b, we need another equation.
From the given table, we know that the sum of probabilities should equal 1. So we can create another equation:
0.1 + 0.2 + a + b + 0.2 = 1
Simplifying this equation:
a + b = 1 - 0.1 - 0.2 - 0.2
a + b = 0.5
Now we have a system of linear equations:
3a + 4b = 1.6
a + b = 0.5
We can solve this system of equations using various methods such as substitution or elimination. I will use the elimination method here:
Multiply the second equation by 4 to match the coefficients of b:
4a + 4b = 2
Now subtract this equation from the first equation:
(3a + 4b) - (4a + 4b) = 1.6 - 2
Simplifying:
-1a = -0.4
Divide both sides by -1:
a = 0.4
Now substitute the value of a into the second equation to solve for b:
0.4 + b = 0.5
Subtract 0.4 from both sides:
b = 0.5 - 0.4
b = 0.1
Therefore, the value of a is 0.4 and the value of b is 0.1.