A random variable Y has the following CDF ; Fy(y)=
0 x<-1
0.25 -1≤x<2
0.5 2≤x<3
1 x≥3
a)Find P[y\B] (Y) FOR B= y<3
b) Find E [y\B]
c)Find Var [y\B]
To find the probability function, expected value, and variance of a random variable Y given an event B, we need to use the conditional probability and conditional expected value formulas. Here's how you can calculate them:
a) P[Y|B] for B = Y < 3:
To find the probability that Y is less than 3 given the event B, we need to calculate P[Y < 3 | Y < 3]. Since Y < 3 is a subset of Y < 3, the probability will be 1.
P[Y|B] = P[Y < 3 | Y < 3] = 1
b) E[Y|B]:
To find the expected value of Y given the event B, we need to calculate the conditional expected value E[Y|B] using the conditional probability distribution.
E[Y|B] = ∑ y * P[Y = y | B]
In this case, we have two different values for Y when B = Y < 3. Using the conditional probability distribution, we get:
E[Y|B] = (-1) * P[Y = -1 | Y < 3] + (2) * P[Y = 2 | Y < 3]
Using the given cumulative distribution function (CDF), we can calculate the probabilities:
P[Y = -1 | Y < 3] = P[Y = -1] / P[Y < 3] = 0.25 / 0.5 = 0.5
P[Y = 2 | Y < 3] = P[Y = 2] / P[Y < 3] = (0.5 - 0.25) / 0.5 = 0.25 / 0.5 = 0.5
Substituting these values into the formula, we get:
E[Y|B] = (-1) * 0.5 + (2) * 0.5 = 1
Therefore, E[Y|B] = 1.
c) Var[Y|B]:
To calculate the variance of Y given the event B, we use the formula:
Var[Y|B] = E[(Y - E[Y|B])^2 | B]
For this calculation, we need to find E[Y|B] from part (b) and calculate E[(Y - E[Y|B])^2 | B].
Using the given CDF, we have two cases:
1. Y = -1 with probability 0.25
2. Y = 2 with probability 0.25
Therefore,
Var[Y|B] = (-1 - 1)^2 * P[Y = -1 | Y < 3] + (2 - 1)^2 * P[Y = 2 | Y < 3]
Plugging in the probabilities we calculated earlier, we get:
Var[Y|B] = 4 * 0.5 + 1 * 0.5 = 2 + 0.5 = 2.5
Therefore, Var[Y|B] = 2.5.