Question

1.Assume Y is a discrete random variable with a mean and a variance of
2, and let X = Y + 1.

a) Do you expect the mean of X to be larger than, smaller than, or
equal to µ = 𝐸(𝑌 )? Why?

b) Express 𝐸(𝑋) = 𝐸(𝑌 + 1) in terms of µ = E(Y ). Does
this result agree with your answer to part (a)?

c) Recalling that the variance is a measure of spread or dispersion,
do you expect the variance of X to be larger than, smaller than, or
equal to 𝜎 2 = 𝑉(𝑌 )? Why?

2. In your notes calculate the median and interquartile for Example
3.2.2 and 3.2.3 under Continuous Uniform Distribution notes.

3. Work out the mean and variance of Gamma Distribution.

Accepted Answer

1. a)  x_i=y_i+1   ud835 udc38( ud835 udc4c )= sum y_i /n   ud835 udc38(X )= sum (y_i+1) /n  so,  ud835 udc38(X )> ud835 udc38(Y )  b)  ud835 udc38( ud835 udc4b) = ud835 udc38( ud835 udc4c + 1)= ud835 udc38( ud835 udc4c)+E(1)= ud835 udc38( ud835 udc4c)+1  result agree with answer to part (a) c)  ud835 udc49( ud835 udc4c )= frac{ sum (y_i-E(Y))^2}{n}   ud835 udc49(X )= frac{ sum (x_i-E(X))^2}{n}= frac{ sum (y_i+1-E(Y)-1)^2}{n}= frac{ sum (y_i-E(Y))^2}{n}  so,  ud835 udc49(X )= ud835 udc49(Y )  2. for Continuous Uniform Distribution with pdf  f(x)= frac{1}{b-a}  median is  m= frac{a+b}{2}  interquartile:  q_3-q_1=a+3(b-a) /4-a-(b-a) /4=(b-a) /2  3. for Gamma Distribution with pdf  f()= frac{ lambda ^{ alpha}x^{ alpha -1}e^{- lambda x}}{ Gamma( alpha)}  where gamma function   Gamma( alpha)= int^{ infin}_0 x^{ alpha -1}e^{-x}dx  mean:  E(X)= alpha / lambda  variance:  Var(X)= alpha / lambda^2

Question #229744

Expert's answer

Answer in progress...