What is the characteristic function of a random variable?
What is the characteristic function of a random variable?
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.
What is known as double exponential distribution?
Laplace distribution, or bilateral exponential distribution, consisting of two exponential distributions glued together on each side of a threshold. Gumbel distribution, the cumulative distribution function of which is an iterated exponential function (the exponential of an exponential function).
What is the distribution of the sum of two exponential random variables?
The answer is a sum of independent exponentially distributed random variables, which is an Erlang(n, λ) distribution. The Erlang distribution is a special case of the Gamma distribution. The difference between Erlang and Gamma is that in a Gamma distribution, n can be a non-integer.
Why Laplace distribution is called double exponential distribution?
It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back, although the term is also sometimes used to refer to the Gumbel distribution.
How do you find the characteristic function?
The characteristic function has similar properties to the MGF. For example, if X and Y are independent ϕX+Y(ω)=E[ejω(X+Y)]=E[ejωXejωY]=E[ejωX]E[ejωY](since X and Y are independent)=ϕX(ω)ϕY(ω). More generally, if X1, X2., Xn are n independent random variables, then ϕX1+X2+⋯+Xn(ω)=ϕX1(ω)ϕX2(ω)⋯ϕXn(ω).
What is the characteristic function used for?
The use of the characteristic function is almost identical to that of the moment generating function: it can be used to easily derive the moments of a random variable; it uniquely determines its associated probability distribution; it is often used to prove that two distributions are equal.
How do you find a double exponential function?
How to Calculate the Double Exponential Moving Average
- Choose any lookback period, such as five periods, 15 periods, or 100 periods.
- Calculate the EMA for that period. This is EMA(n).
- Apply an EMA with the same lookback period to EMA(n).
- Multiply two times the EMA(n) and subtract the smoothed EMA.
How do you add two exponential functions?
The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents. The third law states that in order to raise a power to a new power, we multiply the exponents.
What is the sum of exponential random variables?
The sum of n exponential (β) random variables is a gamma (n, β) random variable. Since n is an integer, the gamma distribution is also a Erlang distribution. The sum of the squares of N standard normal random variables has a chi-squared distribution with N degrees of freedom.
What is the characteristic function of Laplace distribution?
Now, the characteristic function is φX(u)=E[eiuX]=∫∞−∞eiux⋅12e−|x|dx=12∫0−∞eiux⋅e−(−x)dx+12∫∞0eiux⋅e−xdx=12∫0−∞(cos(ux)+isin(ux))⋅exdx+12∫∞0(cos(ux)+isin(ux))⋅e−xdx=12∫∞0(cos(ux)−isin(ux))⋅e−xdx+12∫∞0(cos(ux)+isin(ux))⋅e−xdx=∫∞0cos(ux)⋅e−xdx=−e−xu2+1(cos(ux)−usin(ux))|∞0(1)=0−(−1u2+1(1−0))=1u2+1.
What is MLE of Laplace distribution?
Maximum likelihood estimators (MLE’s) are presented for the parame- ters of a univariate asymmetric Laplace distribution for all possible situations related to known or unknown parameters. These estimators admit explicit form in all but two cases.