Question

Suppose (S,P) is a binomial distribution with n trials and probability of success p. Let X be he random variable X(k)=k³, where k is the number of successes. Calculate E[X].

Answer 1

To calculate the expected value of the random variable X in a binomial distribution, use the formula µ = np, where µ is the mean, n is the number of trials, and p is the probability of success. Multiply the mean of X(k) by the corresponding probability of getting k successes to calculate E[X].

Step-by-step explanation:

To calculate the expected value of the random variable X, we need to use the formula µ = np, where µ represents the mean, n represents the number of trials, and p represents the probability of success. In this case, X(k) = k³, where k is the number of successes. So, to calculate E[X], we need to find the mean of X(k) multiplied by the probability of getting k successes. Let's use an example to illustrate this:

If we have a binomial distribution with n = 10 trials and p = 0.5, the probabilities of getting 0, 1, 2, 3, ..., 10 successes would be:

P(X=0) = (0)³ * (0.5)^0 * (1-0.5)^(10-0) = 0.0009765625

P(X=1) = (1)³ * (0.5)^1 * (1-0.5)^(10-1) = 0.009765625

P(X=2) = (2)³ * (0.5)^2 * (1-0.5)^(10-2) = 0.0439453125

P(X=10) = (10)³ * (0.5)^10 * (1-0.5)^(10-10) = 0.0009765625

To find E[X], we calculate the sum of (k³ * P(X=k)) for all possible values of k:

E[X] = (0³ * 0.0009765625) + (1³ * 0.009765625) + (2³ * 0.0439453125) + ... + (10³ * 0.0009765625)

By calculating this sum, we can find the expected value of X for any given binomial distribution.