Question

Consider a random variable X that takes on the following values with the following probabilities: P(X=1)=0.2,P(X=2)=0.1,P(X=5)=0.4,P(X=8)=0.3 a) Verify that this is a proper probability distribution b) Find the mean μ for X c) Find the standard deviation σ for X

Answer 1

The probability distribution is proper. The mean (μ) for X is 6.1. The standard deviation (σ) for X is approximately 2.89.

Step-by-step explanation:

a. Probability Distribution:
To determine if this is a proper probability distribution, we need to check if the probabilities add up to 1. Let's calculate:

P(X=1) + P(X=2) + P(X=5) + P(X=8) = 0.2 + 0.1 + 0.4 + 0.3 = 1

Since the sum of the probabilities equals 1, this is a proper probability distribution.

b. Mean (μ) of X:
To find the mean, we multiply each X value by its corresponding probability and add them up. Let's calculate:

μ = (1*0.2) + (2*0.1) + (5*0.4) + (8*0.3) = 1.5 + 0.2 + 2 + 2.4 = 6.1

Therefore, the mean μ for X is 6.1.

c. Standard Deviation (σ) of X:
The formula for calculating standard deviation is σ = sqrt(Σ(X-μ)^2*P(X)). Let's calculate:

σ = sqrt((1-6.1)^2*0.2 + (2-6.1)^2*0.1 + (5-6.1)^2*0.4 + (8-6.1)^2*0.3) = sqrt(25.61*0.2 + 18.81*0.1 + 1.21*0.4 + 2.89*0.3) ≈ sqrt(5.122 + 1.881 + 0.484 + 0.867) ≈ sqrt(8.354) ≈ 2.89

Therefore, the standard deviation σ for X is approximately 2.89.