1. Use the following to answer questions: The table below presents the probability distribution for X = the number of email messages you will receive in the next 10 minutes. (9 pts)
X 0 1 2 3
P(X) .35 .35 .25 .05
a. What is the probability that no emails will be received?
b. What is the probability that at least one email will be received?
c. What is E(X), the expected number of emails that will be received?
d. What is Var(X), the variance of the number of emails that will be received?
e. What is the Std(X), the standard deviation of the number of emails that will be received?
I will do one:
c: expected value: 0*.35+1*.35 + 2*.25 + 3*.05= 1
To answer these questions, we need to understand the probability distribution for X, which represents the number of email messages you will receive in the next 10 minutes. The table provides the probabilities for different values of X.
a. To find the probability that no emails will be received (X = 0), we look at the corresponding probability in the table, which is P(X=0) = 0.35.
b. To find the probability that at least one email will be received, we need to calculate the sum of probabilities for X values greater than or equal to 1. In this case, we add the probabilities for X=1, X=2, and X=3. Therefore, P(X>=1) = P(X=1) + P(X=2) + P(X=3) = 0.35 + 0.25 + 0.05 = 0.65.
c. To find the expected number of emails that will be received (E(X)), we multiply each value of X by its corresponding probability and then sum them up. In this case, we multiply X= 0 * 0.35 + 1 * 0.35 + 2 * 0.25 + 3 * 0.05. Thus, E(X) = 0 * 0.35 + 1 * 0.35 + 2 * 0.25 + 3 * 0.05 = 0 + 0.35 + 0.5 + 0.15 = 1.
d. To find the variance of the number of emails that will be received (Var(X)), we need to calculate the squared difference between each X value and the expected value (E(X)), multiply it by its corresponding probability, and then sum them up. In this case, we calculate (0-1)^2 * 0.35 + (1-1)^2 * 0.35 + (2-1)^2 * 0.25 + (3-1)^2 * 0.05. Therefore, Var(X) = (0-1)^2 * 0.35 + (1-1)^2 * 0.35 + (2-1)^2 * 0.25 + (3-1)^2 * 0.05 = 1^2 * 0.35 + 0^2 * 0.35 + 1^2 * 0.25 + 2^2 * 0.05 = 0.35 + 0 + 0.25 + 0.2 = 0.85.
e. To find the standard deviation of the number of emails that will be received (Std(X)), we take the square root of the variance (Var(X)). Therefore, Std(X) = √0.85 = 0.92 (rounded to two decimal places).