Probability and Variance-Bernoulli, Binomial, Negative BinomialI am taking a course in probability and I have trouble computing the variance of a random variable.There are 2 cases we saw in class that I would like to understand:First I will state the definition of variance of expected value:- If X is a random variable who's values are in N, then E ( X ) = Σ n ≥ 0 P ( X &gt; n )- If X is a random variable and E(X) exists, then the variance of X is: V a r ( X ) = E ( ( X − E ( X ) ) 2 ) Now here are the examples I'd like to understand:1. binomial distribution: If X is a random variable that follows a binomial distribution of parametres n and p then we can write X = X 1 + X 2 + X 3 + . . . + X n where the X i 's are bernoulli variables of parametre p. Then V a r ( X ) = n p ( 1 − p )2. negative binomial distribution (Pascal law): If X is a random variable that follows a pascal law of parameters r and p the X + r = X 1 + . . . + X r where X i 's are independant geometric variables. Then V a r ( X ) = V a r ( X + r ) = r 1 − p p 2

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Probability and Variance-Bernoulli, Binomial, Negative BinomialI am taking a course in probability and I have trouble computing the variance of a random variable.There are 2 cases we saw in class that I would like to understand:First I will state the definition of variance of expected value:- If X is a random variable who&#039;s values are in N, then      E    (  X  )  =      Σ          n      ≥      0            P    (  X  &amp;gt;  n  )- If X is a random variable and E(X) exists, then the variance of X is:  V  a  r  (  X  )  =      E              (        (  X  −      E    (  X  )      )    2              )      Now here are the examples I&#039;d like to understand:1. binomial distribution: If X is a random variable that follows a binomial distribution of parametres n and p then we can write   X  =      X    1    +      X    2    +      X    3    +  .  .  .  +      X    n   where the       X    i  &#039;s are bernoulli variables of parametre p. Then  V  a  r  (  X  )  =  n  p  (  1  −  p  )2. negative binomial distribution (Pascal law): If X is a random variable that follows a pascal law of parameters r and p the   X  +  r  =      X    1    +  .  .  .  +      X    r   where       X    i  &#039;s are independant geometric variables. Then  V  a  r  (  X  )  =  V  a  r  (  X  +  r  )  =  r                    1        −        p                    p        2

Harper Brewer · Accepted Answer

Step 1If X is Bernoulli, with   P  (  Success  )  =  p  ,, then   E  (  X  )  =  (  0  )  (  1  −  p  )  +  1  p  =  p  .. Because       0    2    =  0 and       1    2    =  1, we also have   E  (      X    2    )  =  p  ..Binomial: Then for   Y  ∼  B  i  n  o  m  (  n  ,  p  )  ,, we have  E  (  Y  )  =  E      (          ∑              i        =        1            n              X      i        )    =      ∑          i      =      1        n    E  (      X    i    )  =  n  p  .Also, direct proofs, with  E  (  Y  )  =      ∑          i      =      0        n    i                              (                            n        i                              )                          p    i    (  1  −  p      )          n      −      i        =      ∑          i      =      1        n    i                              (                            n        i                              )                          p    i    (  1  −  p      )          n      −      i        =  ⋯  =  n  p  (  1  )  =  n  p  ,(with a change in index   j  =  i  −  1) are given in many elementary texts.Step 2Similarly, using independence,  V  a  r  (  Y  )  =  V  a  r      (          ∑              i        =        1            n              X      i        )    =      ∑          i      =      1        n    V  a  r  (      X    i    )  =  n  p  (  1  −  p  )  .Again here, proofs using about the same method as above to find   E  (  Y  (  Y  −  1  )  )  ,, and then Var(Y) from that and E(Y), are given in many elementary texts.For the negative binomial, it seems intuitive that the avarage waiting time for the rth Success should be r/p. A rigorous derivation of the expectation of a negative binomial random variable often uses some sort of trick involving differentiation of a sum. Probably the easiest route is to find the moment generating function and differentiate it to get the mean and variance.

Probability and Variance-Bernoulli, Binomial, Negative Binomial. First I will state the definition of variance of expected value: If X is a random variable who's values are in N, then E(X)=sum_{n >= 0}P(X>n)

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