# Question

Use the Maclaurin’s series expansion of the moment-generating function of the standard normal distribution to show that

(a) µr = 0 when r is odd;

(b) µr = r! 2r / 2r/2(r/2)! when r is even.

(a) µr = 0 when r is odd;

(b) µr = r! 2r / 2r/2(r/2)! when r is even.

## Answer to relevant Questions

If we let KX(t) = lnMX – µ(t), the coefficient of tr/r! in the Maclaurin’s series of KX(t) is called the rth cumulant, and it is denoted by kr. Equating coefficients of like powers, show that (a) k2 = µ2; (b) k3 = ...To prove Theorem 6.10, show that if X and Y have a bivariate normal distribution, then (a) Their independence implies that ρ = 0; (b) ρ = 0 implies that they are independent. Theorem 6.10 If two random variables have a ...A point D is chosen on the line AB, whose midpoint is C and whose length is α. If X, the distance from D to A, is a random variable having the uniform density with α = 0 and β = α, what is the probability that AD, BD, ...A certain kind of appliance requires repairs on the average once every 2 years. Assuming that the times between repairs are exponentially distributed, what is the probability that such an appliance will work at least 3 years ...(a) Use a computer program to find the probability that a random variable having the normal distribution with the mean –1.786 and the standard deviation 1.0416 will assume a value between – 2.159 and 0.5670. (b) ...Post your question

0