The question is incomplete and the complete question is;
Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine the mean, variance, and standard deviation of the random variable Y = 5X and compare to the corresponding results for X.
Answer:
For random variable X;
Mean = 4.5
Variance = 8.25
Standard deviation = 2.87
For random variable 5X;
Mean = 22.5
Variance = 206.25
Standard Deviation = 14.36
Comparing these values of variable 5X to variable X, we can say the following ;
- The mean of variable 5X is five times the mean of Variable X
- The variance of variable 5X is 25 times the variance of variable X.
- The standard deviation of variable 5X is five times the standard deviation of Variable X
Explanation:
From the question, the random variable X has the parameters of;
a=0 and b=9
Thus for the mean; E(X) = (0+9)/2 = 9/2 = 4.5
Variance;
Thus Var(X) = [(9 + 0 + 1)² - 1]/12 = 99/12 = 8.25
We know that;
standard deviation = √(Var(X))
Thus, Standard deviation = √(8.25) = 2.87
Now let's do the same for the random variable 5X;
E(5X) = 5 x (E(X) = 5 x 4.5 = 22.5
Var(5X) = E[(5X - E(5X))²]
Factorizing out 5,we get
Var(5X) = 5² x E [(X - E(X))²]
Thus Var(5X) = 25 x Var(X) = 25 x 8.25 = 206.25
Standard deviation of random variable 5X = √Var(5X) = √206.25 = 14.36