E[Xi]= 0.05
var(Xi)= 0.0475
Give numerical answers for parts (1) and (2).
E[Xi]= ?
var(Xi)= ?
Let A be the event that the manhole cover did not explode yesterday (i.e., X0=0). Find the conditional PDF of Q given A. Express your answer in terms of q using standard notation.
For 0≤q≤0.1, fQ∣A(q)= ?
var(Xi)= 0.0475
(1-q)/0.095
enjoy the 6.041 from MIT, guys ^_^
(1) Expected Value (E[Xi]):
The expected value of a Bernoulli random variable is the probability of a success, which is denoted by p. In this case, p is the unknown probability of the manhole cover exploding on any given day. However, since Q is uniformly distributed between 0 and 0.1, we can consider p as the average of this range.
Therefore, E[Xi] = p = (0 + 0.1) / 2 = 0.05.
Hence, the expected value of Xi is 0.05.
(2) Variance (var(Xi)):
The variance of a Bernoulli random variable is calculated by multiplying the probability of success, p, by the probability of failure, 1-p.
In this case, since p = 0.05, the probability of failure is 1 - p = 1 - 0.05 = 0.95.
var(Xi) = p * (1 - p) = 0.05 * 0.95 = 0.0475.
Therefore, the variance of Xi is 0.0475.
(3) Conditional Probability (fQ|A(q)):
To find the conditional PDF of Q given the event A, we need to determine the probability distribution of Q when we know that X0 = 0 (the manhole cover did not explode yesterday).
Given X0 = 0, the probability that Q falls in the interval [q, q + dq] is the same as the probability that Q is smaller than q + dq, but greater than or equal to 0.
Since Q is uniformly distributed between 0 and 0.1, the probability that Q is smaller than q + dq is (q + dq) / 0.1.
On the other hand, the probability that Q is smaller than 0 (which should be impossible since Q is always positive) is 0. Hence, the probability distribution of Q given A is 0 for q < 0.
Therefore, the conditional PDF of Q given A, fQ|A(q), is given as:
fQ|A(q) = [(q + dq) / 0.1] * 1(q ≥ 0),
where 1(q ≥ 0) is the indicator function that is 1 when q ≥ 0, and 0 otherwise.
Note: It is important to remember that this is a simplified model assuming uniform distribution for Q and independence between the manhole explosions on different days. In reality, other factors may influence the probability of manhole cover explosions.