Verified answer

Answer:

To calculate the variance and standard deviation of a discrete random variable, we first need to find the expected value.

The expected value E(X) of a discrete random variable X is defined as:

E(X) = Σ x P(X=x)

where the sum is taken over all possible values of X, and P(X=x) is the probability that X takes the value x.

Using the given probabilities, we can calculate the expected value as:

E(X) = (0 x 0.75) + (1 x 0.17) + (2 x 0.04) + (3 x 0.025) + (4 x 0.01) + (5 x 0.005) = 0.485

Now, we can calculate the variance of X using the formula:

Var(X) = E(X^2) - [E(X)]^2

where E(X^2) is the expected value of X^2.

Using the probabilities, we can calculate E(X^2) as:

E(X^2) = (0^2 x 0.75) + (1^2 x 0.17) + (2^2 x 0.04) + (3^2 x 0.025) + (4^2 x 0.01) + (5^2 x 0.005) = 0.7975

Now, we can calculate the variance as:

Var(X) = E(X^2) - [E(X)]^2 = 0.7975 - (0.485)^2 = 0.537275

Finally, we can calculate the standard deviation as the square root of the variance:

SD(X) = sqrt(Var(X)) = sqrt(0.537275) = 0.7335 (approx.)

Therefore, the variance of the discrete random variable is 0.537275, and the standard deviation is approximately 0.7335.