6.041 / 6.431 8. Continuous Random Variables

LECTURE 8 • Readings: Sections 3.1-3.3 Lecture outline • Probability density functions Cumulative distribution functions • Normal random variables • Continuous r.v.’s and pdf’s • A continuous r.v. is described by a probability density function fX Sample Space x fX(x) Event {a < X < b } a b P(a ! X ! b)= ! b a fX (x) dx ! " −" fX (x) dx =1 P(x ! X ! x + !) = ! x+! x fX(s) ds $ fX (x) · ! P(X % B)= ! B fX (x) dx, for “nice” sets B Means and variances • E[X]= ! " −" xfX (x) dx • E[g(X)] = ! " −" g(x)fX(x) dx • var(X)= "2 X = ! " −" (x − E[X])2fX (x) dx • Continuous Uniform r.v. fX (x ) • "2 X = ! b a " x − a + b 2 #2 1 b − a dx = (b − a)2 12 Cumulative distribution function (CDF) FX (x)= P(X ! x) = ! x −" fX(t) dt x fX(x ) a b x CDF a b • Also for discrete r.v.’s: FX (x)= P(X ! x)= $ k!x pX(k) x 1/6 2/6 3/6 1 2 4 x1 2 4 xa b • fX (x)= a ! x ! b • E[X]= Sec. 3.3 Normal Random Variables 155 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 Mixed distributions • Schematic drawing of a combination of a PDF and a PMF x0 1 1/2 0 1/2 • The corresponding CDF: FX (x)= P(X ! x) x11/2 1 CDF 1/4 3/4 Gaussian (normal) PDF • Standard normal N(0, 1): fX (x)= 1 &2# e−x2/2 1 2-1 0 Normal PDF fx(x) x 1 2-1 0 x Normal CDF FX(x) 1 0.5 • E[X] = var(X)=1 • General normal N(µ, "2): fX (x)= 1 "&2# e−(x−µ)2/2"2 • It turns out that: E[X]= µ and Var(X)= "2 . • Let Y = aX + b – Then: E[Y ] = Var(Y )= – Fact: Y ' N(aµ + b, a2"2) Calculating normal probabilities • No closed form available for CDF – but there are tables (for standard normal) • If X ' N(µ, "2), then X − µ " ' N( ) • If X ' N(2, 16): P(X ! 3) = P "X − 2 4 ! 3 − 2 4 # = CDF(0.25) .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .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 The constellation of concepts pX(x) fX (x) FX(x) E[X], var(X) pX,Y (x, y) fX,Y (x, y) pX|Y (x | y) fX|Y (x | y) MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.