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Advanced MathQ&A LibraryLet (S, P) be the sample space in which S = {1, 2, 3, 4} and P is given by P (1) = 0.1, P (2) = 0.2, P (3) = 0.3, and P (4) = 0.4. Find the variance of the following random variables:a. X is the random variable defined by X (1) = X (2) = 1, X (3) = 2, and X (4) = 10.b. Y is the random variable defined by Y (k) = 2k (for k = 1, 2, 3, 4).c. Z is the random variable defined by Z (k) = k2 (for k = 1, 2, 3, 4).Question

Asked Feb 14, 2020

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Let (S, P) be the sample space in which S = {1, 2, 3, 4} and P is given by P (1) = 0.1, P (2) = 0.2, P (3) = 0.3, and P (4) = 0.4. Find the variance of the following random variables:

a. X is the random variable defined by X (1) = X (2) = 1, X (3) = 2, and X (4) = 10.

b. Y is the random variable defined by Y (k) = 2^{k} (for k = 1, 2, 3, 4).

c. Z is the random variable defined by Z (k) = k^{2} (for k = 1, 2, 3, 4).

Step 1

According to the given information, let (S,P) be the sample space in which S={1,2,3,4} and P is given by:

P(1) = 0.1

P(2) = 0.2

P(3) = 0.3

P(4) = 0.4

For part (a)

X is the random variable defined by X(1) = X(2) = 1, X(3) = 2 and X(4) = 10

It is required to calculate the variance of the following random variables.

The expected value is the sum of the product of each possibility x with its probability P(X=x):

Step 2

Now, the variance is the expected value of the square deviation from the mean:

Step 3

For part (b) Y is a random variable defined by Y(k) = 2k

It is required to calculate the variance.

The expected value is the sum of the product of each possibility x with its probability P(Y=y):

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