Suppose you are playing a game. The random variable, Y, represents your net profit in dollars. The probability distribution of Y is as follows. (Round your answers to 4 decimal places, if needed.)
y
-3 -2 -1 3 5
P(Y = y) 0.2 0.19 ? 0.22 0.15
(a) What is the mean for net profit?
(b) What is the median for net profit?
(c) What is the standard deviation for net profit?
To find the missing probability in the probability distribution, we know that the sum of all the probabilities must equal 1.
Sum of probabilities = 0.2 + 0.19 + x + 0.22 + 0.15 = 1
0.76 + x = 1
x = 1 - 0.76
x = 0.24
(a) The mean (or expected value) for net profit can be found by multiplying each net profit value by its corresponding probability, and then summing them up.
Mean = (-3 x 0.2) + (-2 x 0.19) + (-1 x 0.24) + (3 x 0.22) + (5 x 0.15)
Mean = -0.6 - 0.38 - 0.24 + 0.66 + 0.75
Mean = 0.19
Therefore, the mean for net profit is $0.19.
(b) To find the median for net profit, we need to arrange the net profit values in order and find the middle value.
-3, -2, -1, 3, 5
The median is the value in the middle, which is -1.
Therefore, the median for net profit is $-1.
(c) The standard deviation for net profit can be calculated using the following formula:
Standard deviation = sqrt(Sum of [(xi - mean)^2 * P(xi)])
Using the mean value of 0.19:
Standard deviation = sqrt((-3 - 0.19)^2 x 0.2 + (-2 - 0.19)^2 x 0.19 + (-1 - 0.19)^2 x 0.24 + (3 - 0.19)^2 x 0.22 + (5 - 0.19)^2 x 0.15)
Standard deviation = sqrt((13.0191 x 0.2) + (4.3169 x 0.19) + (0.0441 x 0.24) + (6.5791 x 0.22) + (20.7361 x 0.15))
Standard deviation = sqrt(2.60382 + 0.819311 + 0.010584 + 1.447802 + 3.110415)
Standard deviation = sqrt(7.991928)
Standard deviation = 2.8289
Therefore, the standard deviation for net profit is $2.8289.
To determine the missing probability in the probability distribution, we can use the fact that the sum of all probabilities in a probability distribution must equal 1. Therefore, we can subtract the sum of the known probabilities from 1 to find the missing probability.
Let's calculate the missing probability first:
0.2 + 0.19 + ? + 0.22 + 0.15 = 1
0.76 + ? = 1
? = 1 - 0.76
? = 0.24
The missing probability is 0.24.
(a) The mean (or expected value) for net profit can be calculated by multiplying each value of Y by its corresponding probability and summing them up:
Mean = (-3 * 0.2) + (-2 * 0.19) + (-1 * 0.24) + (3 * 0.22) + (5 * 0.15)
Mean = -0.6 - 0.38 - 0.24 + 0.66 + 0.75
Mean = 0.19
Therefore, the mean for net profit is $0.19.
(b) The median for net profit is the middle value when the probabilities are arranged in increasing order.
Arranging the probabilities in increasing order: 0.15, 0.19, 0.22, 0.24.
Since the probabilities do not have unique corresponding values of Y, we can consider the cumulative probabilities to find the median.
Cumulative Probability: 0.15, 0.34, 0.56, 0.8.
The median will have a cumulative probability above 0.5 and below 0.56.
Therefore, the median for net profit is $-2.
(c) The standard deviation can be calculated using the formula:
Standard Deviation = sqrt(sum((Y - mean)^2 * P(Y)))
Using the mean value from part (a):
Standard Deviation = sqrt((-3 - 0.19)^2 * 0.2 + (-2 - 0.19)^2 * 0.19 + (-1 - 0.19)^2 * 0.24 + (3 - 0.19)^2 * 0.22 + (5 - 0.19)^2 * 0.15)
Standard Deviation = sqrt(9.16 * 0.2 + 3.61 * 0.19 + 0.64 * 0.24 + 9.61 * 0.22 + 21.16 * 0.15)
Standard Deviation = sqrt(1.832 + 0.6859 + 0.1536 + 2.1058 + 3.174)
Standard Deviation = sqrt(7.9513)
Standard Deviation â 2.8195
Therefore, the standard deviation for net profit is approximately $2.8195.
To find the mean for net profit (a), we need to multiply each possible profit by its corresponding probability, and then sum up these products.
(a) Mean for net profit:
To find the mean:
Mean = Summation (y * P(Y = y))
Mean = (-3 * 0.2) + (-2 * 0.19) + (-1 * P(Y = -1)) + (3 * 0.22) + (5 * 0.15)
We need to find the missing probability P(Y = -1). To do this, we know that the sum of all probabilities must equal 1.
Sum of probabilities = 0.2 + 0.19 + P(Y = -1) + 0.22 + 0.15 = 1
Rearranging the equation:
P(Y = -1) = 1 - (0.2 + 0.19 + 0.22 + 0.15)
Now we can substitute this value back into the mean equation:
Mean = (-3 * 0.2) + (-2 * 0.19) + (-1 * (1 - (0.2 + 0.19 + 0.22 + 0.15))) + (3 * 0.22) + (5 * 0.15)
Simplifying this expression will give us the mean for net profit.
(b) Median for net profit:
The median is the middle value when the data is arranged in ascending order. To find the median, we need to arrange the profits in ascending order:
-3, -2, -1, 3, 5
Since we have an odd number of values, the median will be the value in the middle. In this case, the median is -1.
(c) Standard deviation for net profit:
To find the standard deviation, we first need to find the variance. The variance is the average of the squared differences between each profit and the mean.
Variance = Summation ((y - mean)^2 * P(Y = y))
Variance = ((-3 - mean)^2 * 0.2) + ((-2 - mean)^2 * 0.19) + ((-1 - mean)^2 * P(Y = -1)) + ((3 - mean)^2 * 0.22) + ((5 - mean)^2 * 0.15)
Next, we need to find the missing probability P(Y = -1) using the same approach as in part (a).
After substituting the value of P(Y = -1) back into the variance equation, we can calculate the variance.
Standard deviation = square root of variance