A random variable x has a following probability distribution.
x: 0 1 2 3
P(X=x) 1/6 1/2 1/5 2/15
How to calculate E(1/(X+1))?
In a production process of a certain item made in large quantities, the proportion of defectives is 0.005. A random sample of 200 is selected. What is the probability that there are 3 defectives in the sample? Assume Poisson distribution. Value of e = 2.71828
To calculate the expected value of the function 1/(X+1), we need to apply the formula for calculating the expected value of a function of a random variable.
The expected value (E) of a function g(X) of a random variable X with probability distribution P(X) is given by the formula:
E[g(X)] = ∑[g(x) * P(X=x)], for all possible values of x
In this case, we have the following function: g(X) = 1/(X+1)
So, we need to calculate E[1/(X+1)].
To do this, we need to multiply each value of X by the probability of that value occurring, and then sum up these products. Let's calculate it step by step:
E[1/(X+1)] = (1/(0+1)) * P(X=0) + (1/(1+1)) * P(X=1) + (1/(2+1)) * P(X=2) + (1/(3+1)) * P(X=3)
Simplifying and substituting the respective probabilities:
E[1/(X+1)] = 1/6 * 1/1 + 1/2 * 1/2 + 1/3 * 1/5 + 1/4 * 2/15
Now, let's calculate the values:
E[1/(X+1)] = 1/6 + 1/4 + 1/15 + 1/30
To add fractions, we need to find a common denominator:
E[1/(X+1)] = 5/30 + 7/30 + 2/30 + 1/30
Now, summing up the numerators:
E[1/(X+1)] = (5 + 7 + 2 + 1)/30
Finally, simplifying:
E[1/(X+1)] = 15/30
Thus, the expected value of the function 1/(X+1) is 1/2.