Math Behind Bell curve ( Normal distribution curve)

Probability distribution is all about how the total probability of 1 is distributed on each outcome in the sample space(all possible outcomes).

Start with flipping a coin, which is a random process Random variable is a function that takes its value based on probability.

Let X1 be a random variable that gets the number of head we obtain when we flip a coin. 

                   So X1 = 0  (tail ,with probability 1/2)

                               = 1 (head,with probability 1/2)

 Here probability is equal for two outcomes. This is called uniform distribution. Similarly rolling a dice, Picking a lucky lot are uniform distribution. If you draw a graph of the possible outcomes and its probability you will get a straight line for uniform distribution. 

 Now let X2 be another random variable that takes number of head of a second coin flip. Compute sum of these two random variable (Both are independent)  and call it as Y.

            Y =X1+X2 = 0 ( both tails TT, probability 1/4 )

                                =1 (with probability 1/2 , HT,TH scenario)

                                =2 (with probability 1/4 HH scenario)

 The distribution for the sum of heads when we flip two coins, is no more equal. Getting one head is more probable than getting two heads or two tails. It will give you a triangular shape when you plot the graph.( when rolling 2 dice , getting sum 7 is more probable than 2 or 12)

 Let Y be sum of 10 such random variables. Each random variable holds number of head on individual coin flips.

 Y = Sum (X1 to X10) 

 It can take values from 0 to 10 .What is the probability of each? We use binomial formula to calculate probability of each outcome (0 head to all 10 heads)

 P(Y=k)=   10 Ck p^k q^(10-k)  (x =0 to 10) 

 This is called binomial distribution. If you calculate the values, you can see that the probability of getting 4,5,6 is more than getting extreme values like all 10 heads or 10 tails. 

 This is because 5 can be obtained in many ways combinatorially. First coin shows head, last 4 heads or First 2 coins head and last 3 heads etc. But for obtaining 10 heads, there is only one way.

 Let Y be total number of heads of 100 coins. Plot graph for 0 to 100 in x axis, and their probability in y axis, you will get Bell shaped Normal curve shape centred somewhere around 50 (here mean =np=100*.5=50).

 Here we started with uniform distribution. You could start with a biased coin which has probability of giving head (say 0.75) than tail(0.25).You will get normal curve shape again centred somewhere around 75 .(Here mean =np = 75).

 Central limit theorem states that whatever may be initial probability distribution , if you repeat experiments for large number of times, your values will be clustered around mean and it will result in Normal curve.

 Assume a stock goes up by 1 % with probability .5  and goes down by 0.5 % with probability 0.5 daily. What would be its return after 100 days (with what probability distribution)? What would be the expected stock price after 100 days(with what probability distribution)?

 

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