Y is a discrete random variable. The table below reports the values of Y and the corresponding frequency, n (the number of observations for each value). Based on this information, what is the expected value of Y?
Y n
50 50
70 100
100 200
200 50
120 100
140 100
150 100
To find the expected value of Y, you need to multiply each value of Y by its corresponding probability and then sum the results.
First, you need to calculate the total number of observations, which is the sum of the frequencies:
Total observations (n) = 50 + 100 + 200 + 50 + 100 + 100 + 100 = 700
Next, calculate the probability of each value by dividing the frequency (n) by the total number of observations:
Probability of Y = n / Total observations
For Y = 50:
Probability of Y = 50 / 700 = 0.0714
For Y = 70:
Probability of Y = 100 / 700 = 0.1429
For Y = 100:
Probability of Y = 200 / 700 = 0.2857
For Y = 200:
Probability of Y = 50 / 700 = 0.0714
For Y = 120:
Probability of Y = 100 / 700 = 0.1429
For Y = 140:
Probability of Y = 100 / 700 = 0.1429
For Y = 150:
Probability of Y = 100 / 700 = 0.1429
Now, calculate the expected value by multiplying each value of Y by its corresponding probability and summing the results:
Expected value of Y = (50 * 0.0714) + (70 * 0.1429) + (100 * 0.2857) + (200 * 0.0714) + (120 * 0.1429) + (140 * 0.1429) + (150 * 0.1429)
Expected value of Y = 3.57 + 10 + 28.57 + 14.29 + 17.14 + 20 + 21.43
Expected value of Y โ 115.00
Therefore, the expected value of Y based on the given information is approximately 115.
To find the expected value of a discrete random variable, you need to multiply each possible value by its corresponding probability and then sum them up.
In this case, you have the values of Y and their corresponding frequencies (n). The probabilities can be calculated by dividing the frequency of each value by the total number of observations.
Let's calculate the expected value step by step:
1. Calculate the total number of observations:
Total number of observations = 50 + 100 + 200 + 50 + 100 + 100 + 100 = 700
2. Calculate the probabilities for each value by dividing the frequency by the total number of observations:
P(Y = 50) = 50/700
P(Y = 70) = 100/700
P(Y = 100) = 200/700
P(Y = 200) = 50/700
P(Y = 120) = 100/700
P(Y = 140) = 100/700
P(Y = 150) = 100/700
3. Multiply each value by its corresponding probability:
(50 * P(Y = 50)) + (70 * P(Y = 70)) + (100 * P(Y = 100)) + (200 * P(Y = 200)) + (120 * P(Y = 120)) + (140 * P(Y = 140)) + (150 * P(Y = 150))
4. Sum up the results:
Expected value of Y = Sum of the products from step 3.
By following these steps and performing the calculations, you can determine the expected value of Y.