When you toss a coin 2n times, the expected number of heads is n. Given the probability density function for a standard normal random variable ϕ ( x ) = 1 2 π e − x 2 2 , approximate the probability that you observe n + j heads when you toss the coin 2n times using the density function of a standard random variable ϕ and the standard deviation σ. Hint: use Stirling's formula: n ! ∼ 2 π n n + 1 2 e − n .I know that the exact probability that I get n + j heads is ( 2 n n + j ) ⋅ 1 2 n and I tried to use Stirling's formula on the binomial but it just made it more complicated.

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When you toss a coin 2n times, the expected number of heads is n. Given the probability density function for a standard normal random variable   ϕ  (  x  )  =            1              2        π                  e          −                        x          2                2            , approximate the probability that you observe   n  +  j heads when you toss the coin 2n times using the density function of a standard random variable   ϕ and the standard deviation   σ. Hint: use Stirling&#039;s formula:   n  !  ∼      2    π        n          n      +              1        2                  e          −      n      .I know that the exact probability that I get   n  +  j heads is                     (                            2        n                    n        +        j                            )              ⋅            1              2        n             and I tried to use Stirling&#039;s formula on the binomial but it just made it more complicated.

Savanah Morton · Accepted Answer

Step 1Let X be the number of heads in 2n tosses of a fair coin. Then X has binomial(2n,1/2) distribution, so   E  (  X  )  =  n and   V  (  X  )  =  n      /    2. To calculate   P  (  X  =  n  +  j  ), use the continuity correction to express   P  (  X  =  n  +  j  ) as                    (1)                              P          (          X          =          n          +          j          )          =          P          (          n          +          j          −                      1            2                    ≤          X          ≤          n          +          j          +                      1            2                    )          .                    Step 2The RHS of (1) is algebraically equal to                    (2)                    P                  (                                                    j                −                                  1                  2                                                            n                                  /                                2                                              ≤                                                    X                −                n                                            n                                  /                                2                                              ≤                                                    j                +                                  1                  2                                                            n                                  /                                2                                              )                .            Step 2We recognize that the middle quantity in (2) has approximately standard normal distribution (by the normal approx to the binomial, i.e., the de Moivre-Laplace theorem). The desired prob is therefore approximately the area under the standard normal density between the two endpoints   L  :=                    j        −                  1          2                            n                  /                2             and   R  :=                    j        +                  1          2                            n                  /                2            . Approximate this area by a rectangle. The rectangle will have height   ϕ  (  z  ), where   z  =            j              n                  /                2             is the midpoint of L and R. The width of the rectangle will be the difference between the two endpoints L and R.

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