LECTURE 8• Readings: Sections 3.1-3.3Lecture outline• Probability density functions• Cumulative distribution functions• Normal random variablesContinuous r.v.’s and pdf’s• A continuous r.v. is described by aprobability density function fXSample Spacex fX(x)Event {a < X < b }a bP(a ≤ X ≤ b)=!bafX(x) dx!∞−∞fX(x) dx =1P(x ≤ X ≤ x + δ) =!x+δxfX(s) ds ≈ fX(x) · δP(X ∈ B)=!BfX(x) dx, for “nice” sets BMeans and variances• E[X]=!∞−∞xfX(x) dx• E[g(X)] =!∞−∞g(x)fX(x) dx• var(X)=σ2X=!∞−∞(x − E[X])2fX(x) dx• Continuous Uniform r.v.x fX (x )ab• fX(x)= a ≤ x ≤ b• E[X]=• σ2X=!ba"x −a + b2#21b − adx =(b − a)212Cumulative distribution function(CDF)FX(x)=P(X ≤ x) =!x−∞fX(t) dtx fX(x )abx CDFab• Also for discrete r.v.’s:FX(x)=P(X ≤ x)=$k≤xpX(k)x 1/62/63/6124x 124Mixed distributions• Schematic drawing of a combination ofa PDF and a PMFx0 11/201/2• The corresponding CDF:FX(x)=P(X ≤ x)x 11/21CDF1/43/4Gaussian (normal) PDF• Standard normal N (0, 1): fX(x)=1√2πe−x2/212-1 0Normal PDF fx(x)x 12-1 0x Normal CDF FX(x)10.5• E[X] = var(X)=1• General normal N(µ, σ2):fX(x)=1σ√2πe−(x−µ)2/2σ2• It turns out that:E[X]=µ and Var(X)=σ2.• Let Y = aX + b– Then: E[Y ] = Var(Y )=– Fact: Y ∼ N (aµ + b, a2σ2)Calculating normal probabilities• No closed form available for CDF– but there are tables(for standard normal)• If X ∼ N (µ, σ2), thenX − µσ∼ N()• If X ∼ N(2, 16):P(X ≤ 3) = P"X − 24≤3 − 24#= CDF(0.25)Sec. 3.3 Normal Random Variables 155.00 .01 .02 .03 .04 .05 .06 .07 .08 .090.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .53590.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .57530.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .61410.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .65170.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .68790.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .72240.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .75490.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .78520.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .81330.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .83891.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .86211.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .88301.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .90151.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .91771.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .93191.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .94411.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .95451.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .96331.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .97061.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .97672.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .98172.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .98572.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .98902.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .99162.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .99362.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .99522.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .99642.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .99742.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .99812.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .99863.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .99903.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .99933.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .99953.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .99973.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998The standard normal table. The entries in this table provide the numerical valuesof Φ(y)=P(Y ≤ y), where Y is a standard normal random variable, for y between 0and 3.49. For example, to find Φ(1.71), we look at the row corresponding to 1.7 andthe column corresponding to 0.01, so that Φ(1.71) = .9564. When y is negative, thevalue of Φ(y) can be found using the formula Φ(y)=1− Φ(−y).The constellation of conceptspX(x) fX(x)FX(x)E[X], var(X)pX,Y(x, y) fX,Y(x, y)pX|Y(x | y) fX|Y(x |
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