Question

I. What is the expected value (mean) of the random variable Y, E(Y), to three significant digits, when Y follows a normal distribution with mean μ = -1.11 and standard deviation σ = 0.39?

II. What is the variance of the random variable Y, Var(Y), to two significant digits, when Y follows a normal distribution with mean μ = -1.11 and standard deviation σ = 0.39?

III. Find the quantile q to three significant digits such that the cumulative distribution function F(Y)(q) equals 0.56, where Y follows a normal distribution with mean μ = -1.11 and standard deviation σ = 0.39.

IV. Assuming the same expectation, find the standard deviation σ to three significant digits such that the cumulative distribution function F(Y)(0.03) equals 0.82, where Y follows a normal distribution with mean μ = -1.11.

Answer 1

The expected value of the random variable Y is -1.11, the variance is 0.1521, the quantile that corresponds to a cumulative distribution of 0.56 is approximately 0.16, and the standard deviation that corresponds to a cumulative distribution of 0.82 is approximately 0.63.

Step-by-step explanation:

I. To calculate the expected value (mean) of a normally distributed random variable Y, we multiply the mean by the probability density function at each point and take the integral over the entire range of possible values. In this case, the mean (μ) is -1.11 and the standard deviation (σ) is 0.39. The expected value can be calculated as:

E(Y) = μ = -1.11

II. The variance of a normally distributed random variable can be calculated by squaring the standard deviation. In this case, the standard deviation (σ) is 0.39. The variance can be calculated as:

0.1521

III. The quantile (q) represents a value below which a certain proportion of the distribution falls. To find the quantile that corresponds to a cumulative distribution of 0.56, we can use a standard normal distribution table or a statistical calculator. In this case, the quantile (q) is approximately 0.16.

IV. Assuming the same expectation, we can use a standard normal distribution table or a statistical calculator to find the standard deviation (σ) that corresponds to a cumulative distribution of 0.82. In this case, the standard deviation (σ) is approximately 0.63.