X f(x)
0 .25
1 .20
2 .15
3 .30
4 .10
Is it prpbsbility distribution valid? Explain and list the requirement for a valid probability distribution
Calculate the expected value x
Calulate the variance of x
calculate the standard deviation of x
To determine if a probability distribution is valid, we need to check if it satisfies the following requirements:
1. The sum of all probabilities must equal 1: This means that when we add up the probabilities of all possible outcomes, the sum should be equal to 1.
Let's check the sum of the probabilities in this case:
Sum = 0.25 + 0.20 + 0.15 + 0.30 + 0.10 = 1
Since the sum of the probabilities in this case is 1, the probability distribution is valid according to the first requirement.
2. All probabilities must be non-negative: Each individual probability must be greater than or equal to 0.
In this case, all the probabilities listed are between 0 and 1, so the second requirement is satisfied.
Now, let's move on to calculating the expected value (mean), variance, and standard deviation of the random variable, x.
To calculate the expected value of x, we need to multiply each value of x by its corresponding probability, and then sum up all the products. In formula form:
Expected Value (E) = Σ(x * P(x))
Using the given data:
E = (0 * 0.25) + (1 * 0.20) + (2 * 0.15) + (3 * 0.30) + (4 * 0.10)
E = 0 + 0.20 + 0.30 + 0.90 + 0.40
E = 1.80
So, the expected value of x is 1.80.
To calculate the variance of x, we need to calculate the squared difference between each value of x and the expected value, multiply it by the probability, and then sum up all the products. In formula form:
Variance (Var) = Σ((x - E)^2 * P(x))
Using the given data:
Var = ((0 - 1.80)^2 * 0.25) + ((1 - 1.80)^2 * 0.20) + ((2 - 1.80)^2 * 0.15) + ((3 - 1.80)^2 * 0.30) + ((4 - 1.80)^2 * 0.10)
Var = (3.24 * 0.25) + (0.16 * 0.20) + (0.04 * 0.15) + (1.44 * 0.30) + (4.84 * 0.10)
Var = 0.81 + 0.032 + 0.006 + 0.432 + 0.484
Var = 1.764
So, the variance of x is 1.764.
Finally, to calculate the standard deviation, we take the square root of the variance:
Standard Deviation (SD) = √Var
Using the calculated variance:
SD = √1.764
SD ≈ 1.33
Therefore, the standard deviation of x is approximately 1.33.