Question

Let Γ(α) be the Gamma function, defined by Γ(α) = ∫∞ 0 e −xx α−1 dx...

Let Γ(α) be the Gamma function, defined by Γ(α) = ∫∞ 0 e −xx α−1 dx for α > 0. Prove that Γ(1/2) = √ π. (Hint: Let y = √ 2x and use properties of the standard normal density function.).

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let 0 <  γ  <  α . Then a 100(1 −  α )% CI for μ...
Let 0 <  γ  <  α . Then a 100(1 −  α )% CI for μ when n is large is Xbar+/-zγ*(s/sqrt(n))The choice γ = α /2 yields the usual interval derived in Section 8.2; if γ ≠ α /2, this confidence interval is not symmetric about . The width of the interval is W=s(zγ+ zα-γ)/sqrt(n). Show that w is minimized for the choice γ = α /2, so that the symmetric interval is the shortest. [ Hints : (a)...
Let X be a gamma random variable with parameters α > 0 and β > 0....
Let X be a gamma random variable with parameters α > 0 and β > 0. Find the probability density function of the random variable Y = 3X − 1 with its support.
Let X be a random variable with probability density function given by f(x) = 2(1 −...
Let X be a random variable with probability density function given by f(x) = 2(1 − x), 0 ≤ x ≤ 1,   0, elsewhere. (a) Find the density function of Y = 1 − 2X, and find E[Y ] and Var[Y ] by using the derived density function. (b) Find E[Y ] and Var[Y ] by the properties of the expectation and the varianc
1. Let the angles of a triangle be α, β, and γ, with opposite sides of...
1. Let the angles of a triangle be α, β, and γ, with opposite sides of length a, b, and c, respectively. Use the Law of Cosines and the Law of Sines to find the remaining parts of the triangle. (Round your answers to one decimal place.) α = 105°;  b = 3;  c = 10 a= β= ____ ° γ= ____ ° 2. Let the angles of a triangle be α, β, and γ, with opposite sides of length a, b,...
let X1, . . . , Xn i.i.d. Gamma(α, β), β > 0 known and α...
let X1, . . . , Xn i.i.d. Gamma(α, β), β > 0 known and α > 0 unknown. (a) Find the sufficient statistic for α (b) Use the sufficient statistic found in (a) to find the MVUE of α n . b) Use the sufficient statistic found in (a) to find the MVUE of α n
Let X = [0, 1) and Y = (0, 2). a. Define a 1-1 function from...
Let X = [0, 1) and Y = (0, 2). a. Define a 1-1 function from X to Y that is NOT onto Y . Prove that it is not onto Y . b. Define a 1-1 function from Y to X that is NOT onto X. Prove that it is not onto X. c. How can we use this to prove that [0, 1) ∼ (0, 2)?
) Let α be a fixed positive real number, α > 0. For a sequence {xn},...
) Let α be a fixed positive real number, α > 0. For a sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that {xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all n). (b) Prove that {xn} is bounded from below. (Hint: use proof by induction to show xn > √ α for all...
Find the value of C > 0 such that the function ?C sin2x, if0≤x≤π, f(x) =...
Find the value of C > 0 such that the function ?C sin2x, if0≤x≤π, f(x) = 0, otherwise is a probability density function. Hint: Remember that sin2 x = 12 (1 − cos 2x). 2. Suppose that a continuous random variable X has probability density function given by the above f(x), where C > 0 is the value you computed in the previous exercise. Compute E(X). Hint: Use integration by parts! 3. Compute E(cos(X)). Hint: Use integration by substitution!
Let F(x) = 1 − e −2x for x > 0 and F(x) = 0 for...
Let F(x) = 1 − e −2x for x > 0 and F(x) = 0 for x ≤ 0. Is F(x) a distribution function? Explain your answer. If it is a distribution function, find its density function.
1. Let D ⊂ C be an open set and let γ be a circle contained...
1. Let D ⊂ C be an open set and let γ be a circle contained in D. Suppose f is holomorphic on D except possibly at a point z0 inside γ. Prove that if f is bounded near z0, then f(z)dz = 0. γ 2. The function f(z) = e1/z has an essential singularity at z = 0. Verify the truth of Picard’s great theorem for f. In other words, show that for any w ∈ C (with possibly...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT