====== Geometric Distribution ====== 기하분포 \begin{align*} \text{Geometric Distribution: } \;\;\; \text{X} & \thicksim Geo(p) \\ p(X = k) & = q^{k-1} \cdot p \\ E\left[ X \right] & = \frac{1}{p} \\ V\left[ X \right] & = \frac{q}{p^2} \\ \\ \end{align*} ====== Geometric Distributions ====== The probability of Chad making a clear run down the slope is 0.2, and he's going to keep on trying until he succeeds. After he’s made his first successful run down the slopes, he’s going to stop snowboarding, and head back to the lodge triumphantly It’s time to exercise your probability skills. The probability of Chad making a successful run down the slopes is 0.2 for any given trial (assume trials are independent). What’s the probability he’ll need two trials? What’s the probability he’ll make a successful run down the slope in one or two trials? Remember, when he’s had his first successful run, he’s going to stop. Hint: You may want to draw a probability tree to help visualize the problem. {{:b:head_first_statistics:pasted:20191030-004453.png?400}} P(X = 1) = P(success in the first trial) = 0.2 P(X = 2) = P(success in the second trial union failure in the first trial) = 0.8 * 0.2 = 0.16 1회 혹은 2회에서 성공할 확률 P(X %%<=%% 2) = P(X = 1) + P(X = 2) = 0.2 + 0.16 = 0.36 {{:b:head_first_statistics:pasted:20191030-005359.png}} | X | P(X=x) | | 1 | 0.2 | | 2 | 0.8 * 0.2 = 0.16 | | 3 | 0.8 * 0.8 * 0.2 = 0.128 | | 4 | 0.8 * 0.8 * 0.8 * 0.2 = 0.1024 | | . . . | . . . . . | | X | P(X=x) | Power of 0.8 | Power of 0.2 | | 1 | 0.80 * 0.2 | 0 | 1 | | 2 | 0.81 * 0.2 | 1 | 1 | | 3 | 0.82 * 0.2 | 2 | 1 | | 4 | 0.83 * 0.2 | 3 | 1 | | 5 | 0.84 * 0.2 | 4 | 1 | | r | . . . . . | r - 1 | 1 | $P(X = r) = 0.8^{r-1} × 0.2$ $P(X = r) = q^{r-1} × p $ This formula is called the **geometric distribution**. * You run a series of independent trials. * (각 시행이 독립적임 = 이번 시행이 이전 시행과 상관없이 일어남) * There can be either a success or failure for each trial, and the probability of success is the same for each trial. (성공/실패만 일어나는 상황에서, 성공하는 확률은 p로 실행햇수동안 일정) * The main thing you’re interested in is how many trials are needed in order to get the first successful outcome. (성공하면 중단하고 성공할 때까지의 확률을 분포로 봄) $ P(X=r) = {p \cdot q^{r-1}} $ $ P(X=r) = {p \cdot (1-p)^{r-1}} $ p = 0.20 n = 29 ## geometric . . . . ## note that it starts with 0 rather than 1 ## since the function uses p * q^(r), ## rather than p * q^(r-1) dgeom(x = 0:n, prob = p) hist(dgeom(x = 0:n, prob = p)) > p = 0.20 > n = 29 > # exact > dgeom(0:n, prob = p) [1] 0.2000000000 0.1600000000 0.1280000000 0.1024000000 0.0819200000 0.0655360000 0.0524288000 [8] 0.0419430400 0.0335544320 0.0268435456 0.0214748365 0.0171798692 0.0137438953 0.0109951163 [15] 0.0087960930 0.0070368744 0.0056294995 0.0045035996 0.0036028797 0.0028823038 0.0023058430 [22] 0.0018446744 0.0014757395 0.0011805916 0.0009444733 0.0007555786 0.0006044629 0.0004835703 [29] 0.0003868563 0.0003094850 > > hist(dgeom(x = 0:n, prob = p)) {{:b:head_first_statistics:pasted:20191030-023820.png}} r번 시도한 이후, 그 이후 어디서든지 간에 성공을 얻을 확률 $$ P(X > r) = q^{r} $$ 예, 20번 시도 후에 어디선가 성공할 확률은? Solution. * 21번째 성공 + 22번째 + 23번째 + . . . . * 위는 구할 수 없음 * 따라서 * 전체 확률이 1이고 20번째까지 성공한 확률을 1에서 빼면 원하는 확률이 됨 * 1 - (1번째 성공 + 2번째 성공 + . . . + 20번째 성공) * 그런데 이것은 * 20번까지는 실패하는 확률 = $q^{r} $ 과 같다 p <- .2 q <- 1-p n <- 19 s <- dgeom(x = 0:n, prob = p) # 20번째까지 성공할 확률을 모두 더한 확률 sum(s) # 따라서 아래는 20번 이후 어디서든지 간에서 성공할 확률 1-sum(s) ## 혹은 (교재가 이야기하는) 20번까지 실패하는 확률 q^20 > p <- .2 > q <- 1-p > n <- 19 > s <- dgeom(x = 0:n, prob = p) > # 20번째까지 성공할 확률 > sum(s) [1] 0.9884708 > # 따라서 아래는 20번 이후 어디서든지 간에서 성공할 확률 > 1-sum(s) [1] 0.01152922 > ## 혹은 (교재가 이야기하는) 20번까지 실패하는 확률 > q^20 [1] 0.01152922 > {{:b:head_first_statistics:pasted:20191031-004954.png}} 그렇다면 r 번 이전에 성공이 있을 확률은? = r 번까지의 실패할 확률의 보수 $$ P(X \le r) = 1 - q^{r} $$ 혹은 1번째 성공 + 2번째 성공 + . . . + r 번째 성공으로 구해도 된다 # r = 20 이라고 하면 p <- .2 q <- 1-p n <- 19 s <- dgeom(x = 0:n, prob = p) sum(s) {{:b:head_first_statistics:pasted:20191031-005143.png}} Note that $$P(X > r) + P(X \le r) = 1 $$ {{:b:head_first_statistics:pasted:20191031-005411.png}} ====== Expected value ====== X가 성공할 확률 p를 가진 Geometric distribution을 따른다 :: $X \sim \text{Geo}(p)$ Reminding . . . [[:b/head_first_statistics/using_discrete_probability_distributions#expected_value_discrete_prob|Expected value in discrete probability]] $E(X) = \sum x*P(X=x)$ | textbook | x | P(X = x) | xP(X = x) | xP(X ≤ x): \\ $E(X) = \sum (x*P(X=x))$ | | r code | trial | ''px <- q^(trial-1)*p'' | ''npx <- trial*(q^(trial-1))*p'' | ''plex <- cumsum(trial*(q^(trial-1))*p)'' | | | | ''px'' | ''npx <- trial*px'' | ''plex <- cumsum(npx)'' | | | | x번째 (trial번째) \\ 성공할 확률 | x번째의 기대치 \\ (주사위 경우처럼) | 그 x번째까지 성공할 \\ 확률에 대한 기대값 | * 우리가 작업하고 있는 채드의 슬로프 타기 예가 얼른 이해가 안된다면 아래 workout의 예를 들어 본다. ^ x ^ p(x) px ^ npx.0 ^ npx = weighted \\ probability at \\ a given spot ^ plex.0 ^ plex ^ ^ | 0 | 0.1 | 0 * 0.1 | 0.00 | 0.00 | 0.00 | | | 1 | 0.15 | 1 * 0.15 | 0.15 | 0.00 + 0.15 | 0.15 | | | 2 | 0.4 | 2 * 0.4 | 0.80 | 0.00 + 0.15 + 0.80 | 0.95 | | | 3 | 0.25 | 3 * 0.25 | 0.75 | 0.00 + 0.15 + 0.80 + 0.75 | 1.7 | | | 4 | 0.1 | 4 * 0.1 | 0.40 | 0.00 + 0.15 + 0.80 + 0.75 + 0.40 | 2.1 | = this is E(x) | * x 일주일에 내가 갈 운동횟수 (workout frequency, 0 to 4) * px 각 횟수에 대한 probability * npx weighted probability * plex cumulative sum of npx (to find out the below last one) * sum of npx = 2.1 = mean of all = expected value of x = E(x) * https://www.youtube.com/watch?v=qafPcWNUiM8 참조 p <- .2 q <- 1-p trial <- c(1:8) px <- q^(trial-1)*p px ## npx <- trial*(q^(trial-1))*p ## 위는 아래와 같음 npx <- trial*px npx ## plex <- cumsum(trial*(q^(trial-1))*p) ## 위는 아래와 같음 plex <- cumsum(npx) plex sumgeod <- data.frame(trial,px,npx,plex) round(sumgeod,3) > p <- .2 > q <- 1-p > trial <- c(1,2,3,4,5,6,7,8) > px <- q^(trial-1)*p > px [1] 0.20000000 0.16000000 0.12800000 0.10240000 0.08192000 0.06553600 0.05242880 0.04194304 > npx <- trial*(q^(trial-1))*p > npx [1] 0.2000000 0.3200000 0.3840000 0.4096000 0.4096000 0.3932160 0.3670016 0.3355443 > plex <- cumsum(trial*(q^(trial-1))*p) > plex [1] 0.200000 0.520000 0.904000 1.313600 1.723200 2.116416 2.483418 2.818962 > sumgeod <- data.frame(trial,px,npx,plex) > round(sumgeod,3) trial px npx plex 1 1 0.200 0.200 0.200 2 2 0.160 0.320 0.520 3 3 0.128 0.384 0.904 4 4 0.102 0.410 1.314 5 5 0.082 0.410 1.723 6 6 0.066 0.393 2.116 7 7 0.052 0.367 2.483 8 8 0.042 0.336 2.819 > * 아래의 예는 위의 workout 예처럼 횟수가 0-4로 정해져 있지 않고 계속 진행됨 (0-무한대) * 하지만 여기서는 100 까지로 한정 (1:100) * 각 지점에서의 probability = geometric probability = q^(trial-1)*p = px * 각 지점에서의 weighted prob = trial * px = npx * 각 단계에서의 기대값을 구하기 위한 누적합계 cumsum(npx) = plex * 아래 그림에서 plex는 각 단계의 probability density를 더해온 값을 말한다. * 그림이 암시하는 것처럼 오른 쪽으로 한 없이 가면서 생기는 그래프의 용적은 기대값이 된다. | {{:b:head_first_statistics:pasted:20251002-072502.png?400}} | | {{:b:head_first_statistics:pasted:20251002-072514.png?400}} | | {{:b:head_first_statistics:pasted:20251002-072522.png?400}} | p <- .2 q <- 1-p trial <- c(1:100) px <- q^(trial-1)*p px npx <- trial*px npx ## plex <- cumsum(trial*(q^(trial-1))*p) ## 위는 아래와 같음 plex <- cumsum(npx) plex sumgeod <- data.frame(trial,px,npx,plex) sumgeod plot(npx, type="l") plot(plex, type="l") > > p <- .2 > q <- 1-p > trial <- c(1:100) > px <- q^(trial-1)*p > px [1] 2.000000e-01 1.600000e-01 1.280000e-01 1.024000e-01 [5] 8.192000e-02 6.553600e-02 5.242880e-02 4.194304e-02 [9] 3.355443e-02 2.684355e-02 2.147484e-02 1.717987e-02 [13] 1.374390e-02 1.099512e-02 8.796093e-03 7.036874e-03 [17] 5.629500e-03 4.503600e-03 3.602880e-03 2.882304e-03 [21] 2.305843e-03 1.844674e-03 1.475740e-03 1.180592e-03 [25] 9.444733e-04 7.555786e-04 6.044629e-04 4.835703e-04 [29] 3.868563e-04 3.094850e-04 2.475880e-04 1.980704e-04 [33] 1.584563e-04 1.267651e-04 1.014120e-04 8.112964e-05 [37] 6.490371e-05 5.192297e-05 4.153837e-05 3.323070e-05 [41] 2.658456e-05 2.126765e-05 1.701412e-05 1.361129e-05 [45] 1.088904e-05 8.711229e-06 6.968983e-06 5.575186e-06 [49] 4.460149e-06 3.568119e-06 2.854495e-06 2.283596e-06 [53] 1.826877e-06 1.461502e-06 1.169201e-06 9.353610e-07 [57] 7.482888e-07 5.986311e-07 4.789049e-07 3.831239e-07 [61] 3.064991e-07 2.451993e-07 1.961594e-07 1.569275e-07 [65] 1.255420e-07 1.004336e-07 8.034690e-08 6.427752e-08 [69] 5.142202e-08 4.113761e-08 3.291009e-08 2.632807e-08 [73] 2.106246e-08 1.684997e-08 1.347997e-08 1.078398e-08 [77] 8.627183e-09 6.901746e-09 5.521397e-09 4.417118e-09 [81] 3.533694e-09 2.826955e-09 2.261564e-09 1.809251e-09 [85] 1.447401e-09 1.157921e-09 9.263367e-10 7.410694e-10 [89] 5.928555e-10 4.742844e-10 3.794275e-10 3.035420e-10 [93] 2.428336e-10 1.942669e-10 1.554135e-10 1.243308e-10 [97] 9.946465e-11 7.957172e-11 6.365737e-11 5.092590e-11 > npx <- trial*px > npx [1] 2.000000e-01 3.200000e-01 3.840000e-01 4.096000e-01 [5] 4.096000e-01 3.932160e-01 3.670016e-01 3.355443e-01 [9] 3.019899e-01 2.684355e-01 2.362232e-01 2.061584e-01 [13] 1.786706e-01 1.539316e-01 1.319414e-01 1.125900e-01 [17] 9.570149e-02 8.106479e-02 6.845471e-02 5.764608e-02 [21] 4.842270e-02 4.058284e-02 3.394201e-02 2.833420e-02 [25] 2.361183e-02 1.964504e-02 1.632050e-02 1.353997e-02 [29] 1.121883e-02 9.284550e-03 7.675228e-03 6.338253e-03 [33] 5.229059e-03 4.310012e-03 3.549422e-03 2.920667e-03 [37] 2.401437e-03 1.973073e-03 1.619997e-03 1.329228e-03 [41] 1.089967e-03 8.932412e-04 7.316071e-04 5.988970e-04 [45] 4.900066e-04 4.007165e-04 3.275422e-04 2.676089e-04 [49] 2.185473e-04 1.784060e-04 1.455793e-04 1.187470e-04 [53] 9.682448e-05 7.892109e-05 6.430607e-05 5.238022e-05 [57] 4.265246e-05 3.472060e-05 2.825539e-05 2.298743e-05 [61] 1.869645e-05 1.520236e-05 1.235804e-05 1.004336e-05 [65] 8.160232e-06 6.628619e-06 5.383242e-06 4.370871e-06 [69] 3.548119e-06 2.879633e-06 2.336616e-06 1.895621e-06 [73] 1.537559e-06 1.246898e-06 1.010998e-06 8.195824e-07 [77] 6.642931e-07 5.383362e-07 4.361904e-07 3.533694e-07 [81] 2.862292e-07 2.318103e-07 1.877098e-07 1.519771e-07 [85] 1.230291e-07 9.958120e-08 8.059129e-08 6.521410e-08 [89] 5.276414e-08 4.268560e-08 3.452790e-08 2.792587e-08 [93] 2.258353e-08 1.826109e-08 1.476428e-08 1.193576e-08 [97] 9.648071e-09 7.798028e-09 6.302080e-09 5.092590e-09 > ## plex <- cumsum(trial*(q^(trial-1))*p) > ## 위는 아래와 같음 > plex <- cumsum(npx) > plex [1] 0.200000 0.520000 0.904000 1.313600 1.723200 2.116416 2.483418 [8] 2.818962 3.120952 3.389387 3.625610 3.831769 4.010440 4.164371 [15] 4.296313 4.408903 4.504604 4.585669 4.654124 4.711770 4.760192 [22] 4.800775 4.834717 4.863051 4.886663 4.906308 4.922629 4.936169 [29] 4.947388 4.956672 4.964347 4.970686 4.975915 4.980225 4.983774 [36] 4.986695 4.989096 4.991069 4.992689 4.994018 4.995108 4.996002 [43] 4.996733 4.997332 4.997822 4.998223 4.998550 4.998818 4.999037 [50] 4.999215 4.999361 4.999479 4.999576 4.999655 4.999719 4.999772 [57] 4.999814 4.999849 4.999877 4.999900 4.999919 4.999934 4.999947 [64] 4.999957 4.999965 4.999971 4.999977 4.999981 4.999985 4.999988 [71] 4.999990 4.999992 4.999993 4.999995 4.999996 4.999997 4.999997 [78] 4.999998 4.999998 4.999998 4.999999 4.999999 4.999999 4.999999 [85] 4.999999 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 [92] 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 [99] 5.000000 5.000000 > sumgeod <- data.frame(trial,px,npx,plex) > sumgeod trial px npx plex 1 1 2.000000e-01 2.000000e-01 0.200000 2 2 1.600000e-01 3.200000e-01 0.520000 3 3 1.280000e-01 3.840000e-01 0.904000 4 4 1.024000e-01 4.096000e-01 1.313600 5 5 8.192000e-02 4.096000e-01 1.723200 6 6 6.553600e-02 3.932160e-01 2.116416 7 7 5.242880e-02 3.670016e-01 2.483418 8 8 4.194304e-02 3.355443e-01 2.818962 9 9 3.355443e-02 3.019899e-01 3.120952 10 10 2.684355e-02 2.684355e-01 3.389387 11 11 2.147484e-02 2.362232e-01 3.625610 12 12 1.717987e-02 2.061584e-01 3.831769 13 13 1.374390e-02 1.786706e-01 4.010440 14 14 1.099512e-02 1.539316e-01 4.164371 15 15 8.796093e-03 1.319414e-01 4.296313 16 16 7.036874e-03 1.125900e-01 4.408903 17 17 5.629500e-03 9.570149e-02 4.504604 18 18 4.503600e-03 8.106479e-02 4.585669 19 19 3.602880e-03 6.845471e-02 4.654124 20 20 2.882304e-03 5.764608e-02 4.711770 21 21 2.305843e-03 4.842270e-02 4.760192 22 22 1.844674e-03 4.058284e-02 4.800775 23 23 1.475740e-03 3.394201e-02 4.834717 24 24 1.180592e-03 2.833420e-02 4.863051 25 25 9.444733e-04 2.361183e-02 4.886663 26 26 7.555786e-04 1.964504e-02 4.906308 27 27 6.044629e-04 1.632050e-02 4.922629 28 28 4.835703e-04 1.353997e-02 4.936169 29 29 3.868563e-04 1.121883e-02 4.947388 30 30 3.094850e-04 9.284550e-03 4.956672 31 31 2.475880e-04 7.675228e-03 4.964347 32 32 1.980704e-04 6.338253e-03 4.970686 33 33 1.584563e-04 5.229059e-03 4.975915 34 34 1.267651e-04 4.310012e-03 4.980225 35 35 1.014120e-04 3.549422e-03 4.983774 36 36 8.112964e-05 2.920667e-03 4.986695 37 37 6.490371e-05 2.401437e-03 4.989096 38 38 5.192297e-05 1.973073e-03 4.991069 39 39 4.153837e-05 1.619997e-03 4.992689 40 40 3.323070e-05 1.329228e-03 4.994018 41 41 2.658456e-05 1.089967e-03 4.995108 42 42 2.126765e-05 8.932412e-04 4.996002 43 43 1.701412e-05 7.316071e-04 4.996733 44 44 1.361129e-05 5.988970e-04 4.997332 45 45 1.088904e-05 4.900066e-04 4.997822 46 46 8.711229e-06 4.007165e-04 4.998223 47 47 6.968983e-06 3.275422e-04 4.998550 48 48 5.575186e-06 2.676089e-04 4.998818 49 49 4.460149e-06 2.185473e-04 4.999037 50 50 3.568119e-06 1.784060e-04 4.999215 51 51 2.854495e-06 1.455793e-04 4.999361 52 52 2.283596e-06 1.187470e-04 4.999479 53 53 1.826877e-06 9.682448e-05 4.999576 54 54 1.461502e-06 7.892109e-05 4.999655 55 55 1.169201e-06 6.430607e-05 4.999719 56 56 9.353610e-07 5.238022e-05 4.999772 57 57 7.482888e-07 4.265246e-05 4.999814 58 58 5.986311e-07 3.472060e-05 4.999849 59 59 4.789049e-07 2.825539e-05 4.999877 60 60 3.831239e-07 2.298743e-05 4.999900 61 61 3.064991e-07 1.869645e-05 4.999919 62 62 2.451993e-07 1.520236e-05 4.999934 63 63 1.961594e-07 1.235804e-05 4.999947 64 64 1.569275e-07 1.004336e-05 4.999957 65 65 1.255420e-07 8.160232e-06 4.999965 66 66 1.004336e-07 6.628619e-06 4.999971 67 67 8.034690e-08 5.383242e-06 4.999977 68 68 6.427752e-08 4.370871e-06 4.999981 69 69 5.142202e-08 3.548119e-06 4.999985 70 70 4.113761e-08 2.879633e-06 4.999988 71 71 3.291009e-08 2.336616e-06 4.999990 72 72 2.632807e-08 1.895621e-06 4.999992 73 73 2.106246e-08 1.537559e-06 4.999993 74 74 1.684997e-08 1.246898e-06 4.999995 75 75 1.347997e-08 1.010998e-06 4.999996 76 76 1.078398e-08 8.195824e-07 4.999997 77 77 8.627183e-09 6.642931e-07 4.999997 78 78 6.901746e-09 5.383362e-07 4.999998 79 79 5.521397e-09 4.361904e-07 4.999998 80 80 4.417118e-09 3.533694e-07 4.999998 81 81 3.533694e-09 2.862292e-07 4.999999 82 82 2.826955e-09 2.318103e-07 4.999999 83 83 2.261564e-09 1.877098e-07 4.999999 84 84 1.809251e-09 1.519771e-07 4.999999 85 85 1.447401e-09 1.230291e-07 4.999999 86 86 1.157921e-09 9.958120e-08 5.000000 ########### 87 87 9.263367e-10 8.059129e-08 5.000000 88 88 7.410694e-10 6.521410e-08 5.000000 89 89 5.928555e-10 5.276414e-08 5.000000 90 90 4.742844e-10 4.268560e-08 5.000000 91 91 3.794275e-10 3.452790e-08 5.000000 92 92 3.035420e-10 2.792587e-08 5.000000 93 93 2.428336e-10 2.258353e-08 5.000000 94 94 1.942669e-10 1.826109e-08 5.000000 95 95 1.554135e-10 1.476428e-08 5.000000 96 96 1.243308e-10 1.193576e-08 5.000000 97 97 9.946465e-11 9.648071e-09 5.000000 98 98 7.957172e-11 7.798028e-09 5.000000 99 99 6.365737e-11 6.302080e-09 5.000000 100 100 5.092590e-11 5.092590e-09 5.000000 > plot(npx, type="l") > plot(plex, type="l") * 기댓값이 86번째 부터는 더이상 늘지 않고 * 계산된 값을 보면 5로 수렴한다. * workout 예처럼 다섯가지의 순서가 있는 것이 아니라서 * 평균을 어떻게 나오나 보기 위해서 100까지 해 봤지만 * 86번째 이후에는 평균값이 더 늘지 않는다 (5에서) * 따라서 위의 geometric distribution에서의 기대값은 5이다. {{:b:head_first_statistics:pasted:20191031-010726.png}} {{:b:head_first_statistics:pasted:20191031-010811.png}} * 그런데 이 기대값은 아래처럼 구할 수 있다. * 위에서 $X \sim \text{Geo}(p)$ 일때, 기대값은 $E(X) = \dfrac{1}{p}$ * 아래는 그 증명이다. ====== Proof of mean and variance of geometric distribution ====== $(4)$, $(5)$에 대한 증명은 [[:Mean and Variance of Geometric Distribution]] ===== e.g., ===== The probability that another snowboarder will make it down the slope without falling over is 0.4. Your job is to play like you’re the snowboarder and work out the following probabilities for your slope success. - The probability that you will be successful on your second attempt, while failing on your first. - The probability that you will be successful in 4 attempts or fewer. - The probability that you will need more than 4 attempts to be successful. - The number of attempts you expect you’ll need to make before being successful. - The variance of the number of attempts. - $P(X = 2) = p * q^{2-1}$ - $P(X \le 4) = 1 - q^{4}$ - $P(X > 4) = q^{4}$ - $E(X) = \displaystyle \frac{1}{p}$ - $Var(X) = \displaystyle \frac{q}{p^{2}}$