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How the beta of a portfolio can equal the market beta if 50 percent of the portfolio is invested in a security that has twice the amount of systematic risk as an average risky security?
The beta of a portfolio is the weighted average of the betas of its individual securities. If 50 percent of the portfolio is invested in a security with a beta of 2 (twice the market's systematic risk), and the other 50 percent is invested in a security with a beta of 0 (no systematic risk), the portfolio's beta can be calculated as follows: (0.5 * 2) + (0.5 * 0) = 1. This means that the portfolio has a beta of 1, equal to the market beta, due to the balancing effect of the low-risk security.
Why the value of correlation coefficient is always between -1 and 1?
Why the value of correlation coefficient is always between -1 and 1?
How gamma are members of the exponential family prove?
Think you've got this backwards. The exponential probability distribution is a gamma probability distribution only when the first parameter, k is set to 1. Consistent with the link below, if random variable X is distributed gamma(k,theta), then for gamma(1, theta), the random variable is distributed exponentially. The gamma function in the denominator is equal to 1 when k=1. The denominator will reduce to theta when k = 1. The first term will be X0 = 1. using t to represent theta, we have f(x,t) = 1/t*exp(-x/t) or we can substitute L = 1/t, and write an equivalent function: f(x;L) = L*exp(-L*x) for x > 0 See: http://en.wikipedia.org/wiki/Gamma_distribution [edit] To the untrained eye the question might seem backwards after a quick google search, yet qouting wikipedia lacks deeper insight in to the question: What the question is referring to is a class of functions that factor into the following form: f(y;theta) = s(y)t(theta)exp[a(y)b(theta)] = exp[a(y)b(theta) + c(theta) + d(y)] where a(y), d(y) are functions only reliant on y and where b(theta) and c(theta) are answers only reliant on theta, an unkown parameter. if a(y) = y, the distribution is said to be in "canonical form" and b(theta) is often called the "natural parameter" So taking the gamma density function, where alpha is a known shape parameter and the parameter of interest is beta, the scale parameter. The density function follows as: f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha) where gamma(alpha) is defined as (alpha - 1)! Hence the gamma-density can be factored as follows: f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha) =exp[alpha*log(beta) + (alpha-1)*log(y) - y*beta - log[gamma(alpha)] from the above expression, the canonical form follows if: a(y) = y b(theta) = -beta c(theta) = alpha*log(beta) d(y) = (alpha - 1)*log(y) - log[gamma(alpha)] which is sufficient to prove that gamma distributions are part of the exponential family.
A stock market analyst estimates that the odds in favor of a stock market going up are 8 to 1. What is the probability of the stock’s not going up in value?
If it has an 8 to 1 chance of going up in value, then there is also a 1 to 8 chance that it won't.
What is the probability of sun rising in east?
On earth it is 1/1 or a certainty
