Let X be a random variable with p.d.f 𝑓(𝑥) = {23 , 𝑤ℎ𝑒𝑛 𝑥 = 113 , 𝑤ℎ𝑒𝑛 𝑥 = 20, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒Find themoment generating function , mean and standard deviation6. Find the m g f of the pdf 𝑓(𝑥) = 𝑞𝑥−1𝑝, 𝑥 = 1,2,3, . . .. and hence evaluate its mean andvariance. (given p+q =1)

Find the m g f of the pdf 𝑓(𝑥) = 𝑞𝑥−1𝑝, 𝑥 = 1,2,3, . . .. and hence evaluate its mean andvariance. (given p+q =1)

Let X be a random variable with p.d.f 𝑓(𝑥) = {23 , 𝑤ℎ𝑒𝑛 𝑥 = 113 , 𝑤ℎ𝑒𝑛 𝑥 = 20, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒Find themoment generating function , mean and standard deviation

1 pointTwo fair dice are rolled simultaneously. Let 𝑌Y represent the sum of the numbers appearing on the upper faces of the dice. Determine the moment-generating function (MGF) of 𝑌−𝐸[𝑌]Y−E[Y]. Select all correct options from below.𝑒−3.5𝑡⋅16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)e −3.5t ⋅ 61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )[𝑒−3.5𝑡⋅16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)]2[e −3.5t ⋅ 61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )] 2 𝑒−7𝑡⋅136(𝑒−5𝑡+2𝑒−4𝑡+3𝑒−3𝑡+4𝑒−2𝑡+5𝑒−𝑡+6+5𝑒𝑡+4𝑒2𝑡+3𝑒3𝑡+2𝑒4𝑡+𝑒5𝑡)e −7t ⋅ 361​ (e −5t +2e −4t +3e −3t +4e −2t +5e −t +6+5e t +4e 2t +3e 3t +2e 4t +e 5t )

A random variable X has moment generating function m(t)=(5+exp(2t))/6. What is the variance of X?

What is the moment generating function (MGF) of a random variable X?