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# Visualizing a binomial distribution

Sal walks through graphing a binomial distribution and connects it back to how to calculate binomial probabilities.

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• Can someone sum up the differences and similarities between discreet, continuous, normal, and binominal distribution. Did I got it right in saying that
1) binomial distribution approaches normal distribution with increase in sample size.
2) discreet distributions generated by random process are binominal distributions. •  Hi there. Let me respond to each of your points in turn.

1) I would agree with you with a couple of clarifications. Let's say you are guessing randomly on a multiple choice test with 10 questions, and each question has 5 answer options with only one correct option. So, your probability of a correct answer is .2, and an incorrect answer is .8. If you Fill out the distribution and graph it, the resulting graph's shape will appear normal, but shifted over towards the left. This is because of the probability involved. Sal's example in this distribution has a .5 probability, which results in a symmetrical distribution, and, as you said, increasing the number of "flips" or events will move the distribution towards a perfect bell curve. If the probability of the event is not .5, then your distribution will be normal but shifted so that it peaks at the mean of the distribution, which can be found using the formula mu=np, where n is the number of events, and p is the probability of success.

2) A discrete distribution generated with randomness will be binomial only if the events are binomial.. So, we must have success, failure, and absolutely nothing in between. We cannot make this distribution for the probability of snowy days, rainy days, and sunny days. We would have to define it so that we are only interested in "sunny" and "not sunny," for example.

I hope that helps out!
• Is it that, Binomial distribution is applicable if the possible outcomes of an experiment is 2? like flipping a coin?
What if it is rolling a dice, where we have 6 outcomes of the experiment, is it Binomial is not applicable in this case? • Do the Bars in the Histogram have anything to do with the Riemann Sum of the Bell Curve? • Yes, they do. You can think of the width of the bars getting narrower and narrower the more possible outcomes there are on the x-axis. For example, if instead of only having the options of one through five, you have one to infinity but it all fits on the x-axis you'd get these really skinny bars and they'd tend toward looking like the bell curve if you get a lot of them, similar to the way that Reimann sums care about minimizing the width of the columns to a really tiny distance and then taking the sum of the areas of the columns.
• Around , is a normal distribution by definition continuous? How does a very large discrete (binomial?) distribution (flip a coin 5 trillion times) differ from a normal distribution? • It doesn't differ significantly beyond the fact that you have discrete results (which can be easily accounted for using a continuity correction). This is the premise of the Central Limit Theorem which states that "the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution" (wikipedia.org)
• What is the difference between the "bell curve" and a Gaussian, to me they look the same,but maybe i'm missing something. :P • Why do we use a histogram if it is a discrete variable? Would we not want to use a bar graph instead? • please i want to know what id P(X<or= x)
or if P(X>or= x) what is the formula an why is that the formula
for both geometric and binomial distributions
and if its difficult to explain with this format, you an write it in paint and take a screenshot with https://snag.gy/ • Why did we use a histogram to represent the distribution? The index is not a continuous variable and there is no range for which the bar represent. It must be a bar chart. • To expand on Victoria's answer, there are a couple more reasons why using a histogram is preferred to visualize the Binomial distribution:

1. The alternative to using a histogram would be to use a line graph. So instead of a bar centered over each value, we would just have a single line at the value. A histogram is generally considered to look better.

2. With bar widths of 1 unit, using the histogram means that the probability of any given value can be visualized by shading that bar and using geometry (`area=base*height`). This provides a nice consistency between discrete and continuous random variables in terms of representing a probability.

3. Sometimes we use continuous distributions to approximate discrete distributions. When we do this, there is a "continuity correction" that gets applied. Using the histogram makes this continuity correction more intuitive.  