I am to find the mean, variance, and standard deviation for the probability distribution. I don't even know where to start. Here is the problem
x P(x)
0 0.19
1 0.32
2 0.28
3 0.21
Formula for the mean:
SUM [x * P(x)]
Take x times P(x) for each value listed. Then add for the mean.
Formula for the variance:
{SUM [x^2 * P(x)] - mean^2}
Square each x. Then take x^2 times P(x) for each value. Add up the values for a total, then subtract mean squared from that total.
For standard deviation:
Take the square root of the variance.
I hope this will help.
To find the mean, variance, and standard deviation for a probability distribution, you need to follow a few steps. Let's go through them one by one.
Step 1: Calculate the Expected Value (Mean)
To find the mean, also known as the expected value (E(X)), you multiply each value of x by its corresponding probability (P(x)), and then sum up all the products. In this case, you have:
E(X) = (0 * 0.19) + (1 * 0.32) + (2 * 0.28) + (3 * 0.21)
E(X) = 0 + 0.32 + 0.56 + 0.63
E(X) = 1.51
So, the mean of the probability distribution is 1.51.
Step 2: Calculate the Variance
To find the variance, you need to calculate the squared deviation of each value of x from the mean, multiplied by its corresponding probability (P(x)). Then, sum up all these products. In this case:
Variance = (0 - 1.51)^2 * 0.19 + (1 - 1.51)^2 * 0.32 + (2 - 1.51)^2 * 0.28 + (3 - 1.51)^2 * 0.21
Variance = 2.8609 * 0.19 + 0.4489 * 0.32 + 0.0009 * 0.28 + 2.6809 * 0.21
Variance = 0.543671 + 0.143648 + 0.000252 + 0.561729
Variance = 1.2493
So, the variance of the probability distribution is 1.2493.
Step 3: Calculate the Standard Deviation
To find the standard deviation, you take the square root of the variance. In this case:
Standard Deviation = √(1.2493)
Standard Deviation ≈ 1.12
So, the standard deviation of the probability distribution is approximately 1.12.