Normal distribution.

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The Normal Distribution

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Presentation transcript:

Normal distribution

Normal distribution X f Are equally distributed random variables common in reality? How does the distribution of a variable observable in nature typically look like?

Normal distribution x = μ  Normal distribution N(μ, σ2) μ , σ f(x) μ x = μ  Normal distribution N(μ, σ2) μ , σ Parameters: x    f(x)  0 x  -  f(x)  0 f(x) maximal for f(x) symmetric around μ In R: dnorm() Density function pnorm() Distribution function qnorm() Quantiles

Normal distribution Example: blood pressure X  N(μ=140, σ2=100) Alteration of μ  translation on the x-axis Alteration σ  dilation of the curve

Normal distribution X  N(μ, σ2) f(x) μ σ Extension below the curve is the total probability (=1).

Normal distribution Question: What is the porbability, that a patient has a blood pressure <= 150 mmHg ? (when blood pressure is normally distributed with μ=140 and σ2=100) P (X ≤ 150) = f ( x ) [mmHg] 120 140 160 180 150

Normal distribution Distribution function (u) of the standardized normal distribution N(0, 1), μ= 0, σ=1  u  .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .5000 .5040 .5080 .5120 .5160 .5190 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7969 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8513 .8554 .8577 .8529 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

Normal distribution N(μ, σ2) N(0, 1) Each normal distribution N(μ, σ2) with arbitrary μ and σ² can be transformed into the standardized normal distribution N(0, 1). N(μ, σ2) Normal distribution N(0, 1) Standardized normal distribution

Normal distribution Central limit theorem: Let X1,X2,…,Xn be independent and identically distributed random variables with E(Xi)=μ and Var(Xi)=σ² für i=1,..,n. Then, the distribution functions of the random variables sn= 1 𝑛 𝑖=1 𝑛 𝑥𝑖−μ σ converge against the distribution function Φ of the standard normal distribution 𝒩(0,1).