# Question: If we let KX t lnMX t the coefficient

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 = µ3;

(c) k4 = µ4 – 3µ22.

(a) k2 = µ2;

(b) k3 = µ3;

(c) k4 = µ4 – 3µ22.

**View Solution:**## Answer to relevant Questions

Show that if a random variable has a uniform density with the parameters α and β, the rth moment about the mean equals (a) 0 when r is odd; (b) 1 / r + 1 (β – α / 2)r when r is even. If the exponent of e of a bivariate normal density is –1/54 (x2 + 4y2 + 2xy+ 2x+ 8y+ 4) Find σ1, σ2, and ρ, given that µ1 = 0 and µ2 = –1. In a certain city, the daily consumption of electric power in millions of kilowatt– hours can be treated as a random variable having a gamma distribution with α = 3 and β = 2. If the power plant of this city has a daily ...If the annual proportion of new restaurants that fail in a given city may be looked upon as a random variable having a beta distribution with α = 1 and β = 4, find (a) The mean of this distribution, that is, the annual ...(a) Use a computer program to find the probability that a random variable having the normal distribution with mean 5.853 and the standard deviation 1.361 will assume a value greater than 8.625. (b) Interpolate in Table III ...Post your question