{ "cells": [ { "cell_type": "markdown", "id": "cac08a26-a424-45b5-8396-e896357c8665", "metadata": {}, "source": [ "2. An electronics manufacturer produces a variety of diodes. Quality\n", "control engineers attempt to insure that faulty diodes will be detected in the\n", "factory before they are shipped. It is estimated that 0.2\\% of the diodes\n", "produced will be faulty.\n", "It is possible to test each diode individually. It is also\n", "possible to place a number of diodes in series and test the\n", "entire group. If this test fails, it means that one or more\n", "of the diodes in that group are faulty. The\n", "estimated testing cost is 4 cents for a single diode, and $3 + n$ cents\n", "for a group of $n > 1$ diodes. If a group test fails, then each diode\n", "in the group must be retested individually.\n" ] }, { "cell_type": "markdown", "id": "f949b1a9-2ebf-4d16-8709-e3626580d201", "metadata": {}, "source": [ "(i) Derive the average cost function $A(n)$ that\n", "should be minimized in this case." ] }, { "cell_type": "markdown", "id": "6958009d-8671-4a81-812d-65bc071323fb", "metadata": {}, "source": [ "Let $D$ be a random variable representing a diode. Thus,\n", "$$\n", " D=\\cases{\\hbox{faulty}& with probability $q$\\cr\n", " \\hbox{good}& with probability $p=1-q$.}\n", "$$\n", "Let $B$ be a random variable representing a test batch of $n$ diodes. Thus,\n", "$$\n", " B=\\cases{\\hbox{all good}& with probability $p^n$\\cr\n", " \\hbox{at least one bad}& with probability $1-p^n$.}\n", "$$\n", "Let $C$ represent the cost of testing a batch of diodes. Thus,\n", "$$\n", " C=\\cases{4& if $n=1$\\cr\n", " 3+n& if $n>1$ and $B=\\hbox{all good}$\\cr\n", " 3+n+4n& if $n>1$ and $B=\\hbox{at least one bad}$.}\n", "$$\n", "Assuming $n>1$ it follows that the expected cost per batch is\n", "$$\\eqalign{\n", " {\\bf E}[C]&=(3+n){\\bf P}(B=\\hbox{all good})+(3+n+4n){\\bf P}(B=\\hbox{at least one bad})\\cr\n", " &=(3+n) p^n + \\big((3+n)+4n\\big)(1-p^n)\n", " %=3+n+4n(1-p^n)\n", " =3+5n-4np^n.\n", "}$$\n", "Therefore, the average per-diode cost function that should be minimized is\n", "$$\n", " A(n)={\\bf E}[C/n]={{\\bf E}[C]\\over n}={3+5n-4np^n\\over n}={3\\over n}+5-4p^n.\n", "$$" ] }, { "cell_type": "markdown", "id": "e1654256-53a5-496b-a5a8-c46a1d5020fd", "metadata": {}, "source": [ "(ii) Find the minimum value of $A(n)$ over the set $n = 1, 2, 3, \\ldots.$" ] }, { "cell_type": "code", "execution_count": 1, "id": "c654e1b4-d962-429a-8c85-23d43f3a6b83", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "66.69890833843708" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q=0.002; p=1-q\n", "log(4/3.5)/log(1/p)" ] }, { "cell_type": "markdown", "id": "c9d7a275-7d28-4b3f-a924-0afe5897f5c6", "metadata": {}, "source": [ "Note that $p^n<3.5/4$ provided $n\\log p<\\log(3.5/4)$ or equivalently when\n", "$$\n", " n>{\\log(3.5/4)\\over\\log p}={\\log(4/3.5)\\over\\log(1/p)}\\approx 66.7.\n", "$$\n", "It follows that $n\\ge 67$ implies\n", "$$\n", " A(n)={3\\over n}+5-4p^n> 5-4(3.5/4)=5-3.5=1.5.\n", "$$\n", "Plotting $A(n)$ over the rangle $[2,68]$ yields" ] }, { "cell_type": "code", "execution_count": 2, "id": "7b0ceeea-4b25-4b1c-b8d3-ae946be4a0e1", "metadata": {}, "outputs": [], "source": [ "using Plots" ] }, { "cell_type": "code", "execution_count": 3, "id": "d6567eab-64f6-471a-9c01-a7927dd5e060", "metadata": {}, "outputs": [ { "data": { "image/png": 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AADpIVkGIlWUAAKCj5BWE6BoFAIAOklUQhmqxsgwAAHSMrIIwSE12kWy4ST0AALhMVkFIRJEBVGmVuggAAPAfcgvCeD0rasRlQgAAcJXalZ3sdvvmzZvz8/Pr6uqGDRs2e/ZsrVZ76W579uz55JNP1Gr1vHnzkpOTnRtbWlqWLVt26NChIUOG3H333RqNxpPlXyJex4rMfKSBefWrAACAbLjUIjx48ODDDz+s0Wj69+//9ttvT5ky5dJ9du/enZOTExsbq9frR40aderUKef2efPm5eXlpaenf/rpp3fddZcna7+ceD0VNXr7iwAAgHwwzq/ekWiz2dRqNWOMiEpLS2NjYy9cuBAfH992n5kzZ/br12/JkiVENH/+/KCgoL/+9a9nz54dOHBgSUlJZGRkRUVFfHz8yZMnExISLvsldDpdS4u78+FfPST+1MRfzVC5+TreYzab9Xq91FV4nSiKFotFp9NJXYjXNTQ0BAcHS12F19lsNofDERgYKHUhXqeQE2q1WtVqtVrtUqeg7LnUItRoNM4UJKKGhgZBEC79QSkoKMjNzXU+zs3N3bp1KxFt3759yJAhkZGRRGQwGAYNGlRYWOix2i8nXkcXzF79CgAAICsdezsgiuKDDz64cOHCsLCwX203mUxRUVHOp0ajsbS0lIhKS0sNBkPrbgaDwbn9shwOx80339z6dOLEiXPnzu1QeURkULPzDaqmpuaOfmKXaWpqEgS5jVG6lLNF2Pr+ScYsFotK5bs9EJ7ibBGKovwnJynkhCqnRajVaq96mB34LnDO58+fb7fbX3755V99SBAEjUZjt9udT1taWgICApwVtG4kIpvNdtlRNq0vcuutt7Y+7du3r/NFOiQpjJc0USc+scvYbDZfLs9TRFEURVEJR9r60y5vgiA4HA4lHKlCTijnXCFB6ErDw9XvAuf8gQceOHHixHfffXfZ6wTdu3cvLi5OTU0louLi4ri4OCLq0aNHUVFR6z6t2y+LMTZz5kwX62lPXDBVNdscTK311UaXSqVSwvtNxphCjlQhh+lsCyrhSBVyQlU/k7oQn+BqXDzxxBP79u375ptv2g5/KC8v37Bhg/PxtGnT8vLyiIhz/tlnn02dOpWIcnNzi4qKDhw4QEQ//vhjaWlpTk6Oh4/gv6kYRXdjpU2YSggAAC5xqUVYUFDw8ssv9+vXrzXGli1blpqaWlhYeO+995aXlxPRo48+Onr06BtuuKGpqammpmb+/PlEFBoa+txzz11//fXZ2dmbNm16/vnnu2A4lnO8TE/5D8wEAAAPcGn6RENDw8mTJ9tu6d+/v16vr6+vLykpGThwoHNjU1NTfn6+Wq0eN25c2072EydOHDlyZNCgQf369WvvS3hq+gQR3b7JMSmB3Z7ko32jmD4hMwoZbY/pEzKjnMEyrnApCLuAB4Nw8S5HRABbPBRBKCUEocwgCGUGQdiWj6aFO7DcKAAAuE6OQajDKmsAAOAqWQYhKzKjRQgAAC6RYxCiaxQAAFwmwyCMCqRGOzXZr74nAACADIOQEfXQsWI0CgEAwAUyDELCeBkAAHCZXIMQ42UAAMAlMg1CPV1AixAAAFwg0yBEixAAAFwjzyBMwAwKAABwjTyDMF5HRWapiwAAAH8gzyBEixAAAFwkzyAM1hAjqvPArSwAAEDm5BmERBSvZxcwXgYAAK5GtkGYoMecegAAuDrZBmG8DpcJAQDg6mQdhOgaBQCAq5FvEKJrFAAAXCDfIESLEAAAXCDjIMRyowAAcHXyDUI9K2nkaBICAMCVyTYIA1UUrKEKi9R1AACAb5NtEBJRPBZaAwCAq5F1EGIqIQAAXI28g5Au4B4UAABwRbIOQj1mUAAAwFXIOQgTdJhTDwAAVyHnIMRgGQAAuCpZByHuUw8AAFcj5yCMDWImK7eLUtcBAAA+TM5BqBbIEMhKLegdBQCAdsk5CAm9owAAcDVyD0KMlwEAgCuSexBiBgUAAFyR7IMQc+oBAOBK5B6EuE89AABckdyDUMcuoEUIAADtk3kQJmCwDAAAXJHMg9DYjepbyOqQug4AAPBVMg9CRhQbxErQKAQAgHbIPAiJKF5PFzBeBgAA2iH/IEzADAoAAGif/IMQMygAAOAKFBCEaBECAED7FBCEmEEBAADtU0AQYrlRAABonxKCEF2jAADQLvkHYXgAOTjV26SuAwAAfJL8g5CIeuhYMS4TAgDA5SgiCHGfegAAaI8yghADRwEAoB3KCEKMlwEAgHYoIwixuAwAALRDGUGI2/MCAEA7FBGEfULoRJ3URQAAgE9SRBD2CmZNdl5ukboOAADwPYoIQkaUGsn2V6F3FAAAfk0RQUhEwyLZPgQhAABcQu36rjU1NQcPHoyIiBg8ePClHz1+/HhxcXHrU0EQcnJyiGjfvn3V1dXOjTqdLisry72CO2lYJFt9HkEIAAC/5moQPvXUU0uXLg0JCcnJycnLy7t0hx9++GHNmjXOx2fOnFGr1SdOnCCixx9/vLKy0mg0ElF8fLxUQTg8ij27T5TkSwMAgC9zNQgffPDBZ5555sUXXzxy5Mhld1i0aNGiRYucj3NyciZOnNj6oaeffvqWW25xs1A39Q9lJguvbaEwrbSFAACAb3H1GmF0dLRGo3Flz7Nnz27dunXOnDmtW44ePfr111+fPn26MwV6iMBoUATGywAAwK914Bqhi5YvX3799dfHx8c7n+r1+sLCwj179uTn5995551vvvlme58oiuILL7zQ+jQzM3Ps2LEeLCw1nPaY7KOjPPiSnWGz2Ww2+d8UShRFhRypcg7T4XCoVCqpC/E65ZxQzjnn8m8bqFQqQbhKk8/DQSiK4kcfffTGG2+0bvnyyy+dRZw7dy41NXX69OnZ2dntfXrrsBoiqqiocDgcHqxtSDjbUsYcDomvFDocDs8el28SRVEhR6qow1TOkUpdhdc5HA7GGGNM6kK87qopSB4Pwu+++665uXnSpEmXFtGrV6/09PSDBw+2F4SCILzyyiueraetjBj+1nFHYKDEFwntdntgYKC0NXQBURQ550o4UpvNpoTDVKlUDodDCUeqkBNKRGq1Wq32fKegP3JrHiHnvLS0tO27p/fff3/u3LmXvZpoNpuPHj3as2dPd76iO5LD2dkG3mSX6usDAIAvcvXtwJYtW1asWOGcFHjvvfdmZ2fPmjWrrq4uNjb22LFjAwYMIKKqqqq1a9fu27ev9bOKi4tnzZo1YcIElUr1+eef9+nTp21jsYtpBRoYxg5W80yj/HsDAADARa4GocFgSEtLS0tLcz51joXR6XQffvhhbGysc2NdXd3y5cuTk5NbP8toND7++OOHDh3inD///PNTpkxxpbvWe4ZFsn2VCEIAAPgF85FRQzabTafTtbS0ePWr/O2o+GMV/8dYKUe+mc1mvV4vYQFdQxRFi8Wi0+mkLsTrGhoagoODpa7C65yjRpVw8UwhJ9RqteIaYSulrDXqNDyK/YiphAAA0IaygnBoBDtWy1uw1BoAAPxMWUHYTU2JwexoDRqFAABwkbKCkIiG435MAADQhuKCcFgU+7ESQQgAABcpLwjRIgQAgDYUF4TDo9iBKu5AFAIAABEpMAhDNNQ9iJ2sQxICAACRAoOQfl5fRuoqAADAJygyCDGtHgAAfqbEIBweiSAEAICLlBiEaVHsxyrfWGIVAACkpsQgjAoknZqda0AUAgCAIoOQiIZHYbwMAAAQKTYIh0USLhMCAAApOAixvgwAABApNgjRNQoAAE4KDcJ4HRM5/dSELAQAUDqFBiERDYtkP1ZJXQQAAEhNuUGI3lEAACAlB+EwrC8DAABKDkK0CAEAgJQchEkhrLaFV1qlrgMAACSl3CBkRKmRbD96RwEAlE25QUiYVg8AAAoPwjHRbONPotRVAACAlBQdhNf1EArLeYNN6joAAEA6ig5CvYYyjezfJWgUAgAol6KDkIgmJQhrL+AyIQCAcik9CKf0ZN8UiSKiEABAqZQehD31zBjIdlUgCQEAFErpQUhEkxLY2gu4TAgAoFAIQpqcIKzBZUIAAKVCEFKmkZVb+LkGZCEAgBIhCElgdF0P4ZsiBCEAgBIhCImIJuMyIQCAUiEIiYiu7yFsxxIzAACKhCAkItJraKSRrccSMwAAyoMgvGgylpgBAFAkBOFFkxPYWiwxAwCgPAjCixKDWVQA240lZgAAFAZB+IvJPdnaIlwmBABQFgThLybFC2vOo0UIAKAsCMJfZEWzkiZ+3owsBABQEAThL1SMbughfIslZgAAlARB+F8mJbA1WGIGAEBJEIT/5fp4oaCMm7HEDACAYiAI/0uIhkYa2Iaf0CgEAFAKBOGv4faEAAA+62wDb/F0U0Xt4dfzf5MS2EsHHSJXCUzqUgAAFM8m0r5Kvt3Et5XzbWWiSmBfXKMaafDkH2gE4a8lhbAwLdtdwTOMSEIAAAlUN1OhiW8rE7eV831VPCmYjYlh03uyVzPUPfWe/8uMILyMO/sK7x0XM4wqqQsBAFCKk3V8eznfVs63l/PiRj7SyEZHs/8dpso0shCNd780gvAy5g8Q+nxmK7eoortJXQoAgEzZRTpQzQvK+LZynl8mqhilRbEx0cKdfYWRRqbtwhEsCMLLCA+gWxKFfxwXnxqGwUQAAB5T10K7K3hBubitnO8w8QQdGxPDJiWwl0aqE4MluxqFILy8RSnCtesc/zNU6Mp3JQAAMsOJjtX+0udZbuGjolmWUfjDUCHDyHS+EUG+UYXvSQlnyeG06qx4exKSEACgA5rstLvCmXxioYmHB7BRRjYqmj0ySEgJZz44IB9B2K6HUlRLfnQgCAEArqqk8eIMh8JyfriGD41gWdHsd/2FZeOEGJ8fbIEgbNdv4tkjO6nQxLMwjwIA4L85OB2svph828p5o52PihZGGdmrGcIIAwv0q0H3CMJ2CYweSBbeOCxm5fjVKQUA8I56GxWW8+3l4nYT32XiPXRsVDSbGMf+OFzoH+rHDYaOBWFNTU1QUFBAQMClH7JYLFartfVpWFgYYxe/L1VVVefOnUtMTIyIiHCn1q73237Cc/tsxY1CD50fn2MAgE4703BxnMu2cn6ugQ+PYqOj2e8HqbKMLOIyUeCXXL0ANn/+/KioqIiIiGXLll12hz/96U+xsbFJPzObzc7tK1as6Nev38MPP9y3b9+8vDzPVN1VgjV0R1/hb0exBjcAKEWLSIUm/toh8ab1ju4rbOPWOtZe4ANC2fKxquo7NJtvVP95hOrGePmkILkehDNnzty5c2d2dvYV9lm0aFH1z4KDg4nIYrEsWrRo1apVW7Zs+eyzzx544IHm5mYPVN2FHkgWlp8ULXap6wAA8BqThVafF/9nl2PMGnvkx7YHtjvOmfmMRLZzqrr4NnVejuqhQUK6gallOnbQ1a7R3NxcImrt7bwsm8127ty52NhYrVbr3LJ+/frw8HBnfObm5ur1+o0bN95www3u1dylkkJYplFYcVq8u79MfwQAQHlETkdqaWeluK2cF5p4hYVnRbMso/BcmjDSwPReXtLM13hysMwHH3zwxRdfmEym++6775VXXhEE4fz58717927dITEx8fz58+19Oud8/fr1rU979+7d9nMl9FCK8Psdjrv6++DsFwAAVzXYaKeJbzfxwnKx0KQyBvJR0XxMNPufIcLAMF+c3tdlPBaEDz/88AsvvCAIwunTp8ePH5+SknLXXXc1NTW1tg6JqFu3bk1NTe29giiKL7zwQuvTa665ZtGiRZ4qzx0jQ0gUtd+cbpoQ7YGLhY2Nje6/iO8TRdFqtXIu/zs7NjY2XrmnRB5sNpvD4bDb5X+RQGYn9HwjK6xgu6qEHRXCGTMNDecZUeJvE/lfh1ijgwS1+mIENMn3z1JgYGDrYbbHY0EYHR3tfJCUlDR79uwtW7bcddddMTEx1dXVrftUVVXFxMS09woqlWrjxo2eqsezfj9Y/McZ1aQkz8yj0Ov1HnkdXyaKokql0ul0UhfidZxzJYSIiBUAABfASURBVJxQZxAGBgZKXYjX+fsJbXbQ3kpeeHFuu6gSLq7qctdANjyKaX6+wmO1Cmq1+qoJoRBe+S4UFxcbDAYiGjFixP79+xsbG3U6ndlsPnjwYFpamje+orfd0Ud4eo/tZJ3Qz5/nygCALJVZaHeFuLeSt13JelpPtiRNnRKOP1lX52oQbty48dSpUyUlJdu2bdNoNBMnTkxMTPzuu+8ee+yxw4cPE9GDDz44cuTIyMjIjRs3fvnll7t27SKi5OTkcePGzZs3795773333XdzcnL69+/vxaPxmgAV/a6/8Ldj4l8zMbkeACTm4HS8ljuTr6CMlzTxdAMbHc0WpQircoVQ7dVfAdpyNQhNJtOZM2emTZtGRGfOnHFOE4yLi5s+fbpzhwEDBqxbt66+vr5v374//vhja+Dl5eW9+OKLb7zxxuDBg5944gkvHEIXWTBQGPqF/X9TVQb5dw4BgM+paaZCEy80idvL+e4KnqBnWUY2Kpo9PEjoHyajq5pSYD4ynMFms+l0upaWFqkLuZJHdjjqWmj5OLcahWaz2a+vQLhIFEWLxaKEa4QNDQ3OWbPyppxrhL5zQp03MNph4tvKeKGJlzTydANz3sMoK5qFudfss1qtuEbYCt+FDng2TZW8yo5luAHASxpstKuCby/nO0xioYlHBLAsI8syskWDhEHhTIU/PN6BIOyAEA0tHSks2ObYM02Nn0gA8Ij/1PEdFbywnG8v56cbeGokG2Vk8wcI748Ton3+BkbygCDsmNuShGUnxL8dFR9MwUIzANAZjXbaU8ELTbzQxAvLxUCVs8OTzesnDIv8ZYYDdBkEYYe9PUo1fq391t54swYArjrTwAvL+Q4TLzTx47V8SATLNLI7+rC/jVLH4eY2UkMQdtjAMPbbfsLjOx0fTcBUCgC4PIud9lQ623y80CSqhYtX+27vIwyPZAH44+FLEISd8cfhquRV9i1lfFwM3soBwEXnGvh2E99h4jtM/EgNHxzBMo1sVhJ7c5Q6Hs0+H4Yg7Aydml7LEBZuc/w4XS3X+5IAwFU12S+uZ1ZYzneYRIGxTCMbFc1m9hbSolggmn1+AkHYSTcnCv84Ib55RHxkMJIQQEFar/btMPGjtb80+97IUifo0ezzSwjCznsrSzVqjX1WEosNwk8/gGy1HeS5wyRqBZZpZFlGdluSMBzNPllAEHZe31B230Dh0Z3ip9n4VQCQD050opbvrOA7TLywnJ+q50MjWaaRzenD3hml7oGrfbKDIHTLH4aqBn1u3/ATz43F7waAH6ttoZ0mvsPEt5Vq9lbbQrUsy8gyjOy3mNunAAhCtwSp6Y0s4d4Cx+6p6vAAqasBAJc5OB2puXipb4eJFzfytCiWZWR3JTk+zgnELGFFQRC6a3KCkF/KZ220f3s91l0D8GllFtppEp3Jt6+Sx+lYhpFlGtnvBwkpP6/k2dAgBiMFFQZB6AEvjVTd+L398Z2O13C3QgBfYnXQvkq+s4I7uz3NNp5hZBlG4YmhQobR3Rs4gGwgCD1AxehfOeqM1fahkeKdfXExAUAynOg/dReTb6eJH63lKeEsw8AmJbAlaULfUHTawGUgCD0jTEtfTVRN+MbeP5Rl4iZNAF2oqpl2mfjOCnFXBd9p4iFalmlkGQZ2O6Y3gGsQhB4zMIy9P049Y4NjxxQVVtEF8J5mB+2v4jsruDP5Kqw83cAyDOy+AcIHuHURdByC0JNujGcLk4Wb1jvyJ6nxPhTAUzjRyTq+q4LvMvFdFfxwDXd2vVwTy55MFQaEMgHvPMENCEIPWzxUOFjN52/FvSkA3FJuoV0VYmv4hQewDCMbaWCzkoRhkSwIf7rAc/DT5GGMaPlY1di19lcPiY9iGVIAl5lttLeS767kO018VwVvtPF0AxtpYIsGqUYamCFQ6vpAvhCEntdNTV9OVGWudvQPZZMS0GUDcHk2kQ5V810VfHcF31XBz5n5kAg20sCm92Ivpgt9QvC7A10EQegV8Tr25UTVlB/sb2apbu2NdiEAEZHI6WQd313Jd1fw3RX8UDVPCmHpBjbSyB5IEQaHM9zUDCSBIPSWkQb27xvUv/neUd1M9w3E7zco1HnzxdjbXcH3VnJDNzbSwNIN7NZEYVgU0+EvEPgA/Bh60eAIlj9Jde06R1Uz/W8qshAUocxCeyr4nkpxdwXfU8nVjKUbWLqBLR4qpBtYBJbkBd+DIPSu3sGsYLL6+nX2Kit/NRNrkYIMVTU7k487/7c6+IgoNsLA5g8Q/mHA3TrBDyAIvS6mG22epJ78g31evmP5WMypAL9X00x7K/neSr6nku+t5NXNPC2KjYhis/uw1zOFXsFIPvAzjHMudQ1ERDabTafTtbS0SF2ItzTZacYGu1pgy9IthjC91OV4nSiKFotFp9NJXYjXNTQ0BAcHS12Fd9U0084y274qfqBWtbeSV1j48CjmDL+0KNYnlMkp+pRwQonIarWq1Wq1Gm0hIgRhV7KJNC/fcaHBvvq6ANlfKUEQ+rWqZtpXyZ3Nvn2VvMLKUyMoNVzM7K5Ji2L95JV8vyLLE3opBGFb+C50HY1AH09QPbKtJfUL+4fjVTm4qT34jHIL7avk+6q48/9qKx8WxdKi2PRebEma0C+UOew2h4MHBmLMF8gQgrBLCYyeT3X8JjFwbr5jVm/25xGqAFw0BClcMPMff469fZVkcfC0KDYsks1IZC+mC30vafM5pCkToCsgCCVwbRw7cJN6/lZHxmr7imxVSjiahuBdzpnsP1b9/K+Sa1U0LJINi2S/7Se8lcUwwgWUDEEojcgA+vwa1fsnxexv7P+bqlo0CKvngydZ7HS4hv9YxfdX8QPV/GA1j+7GnMn3yCBhWBSLwb2KAH6GIJTS7/oJ42PYnM2Ob4vED8arMOMKOq3CSvur+P6fk+9MAx8QyoZGstQINrO3kBrJQrVSlwjgqxCEEksKYVsnqf+8Xxz2pf0PQ1X3DxRw1RCuyi7SyXp+oIofqL74v9VBqRFsaCSbGMceHyKkhDMNxrUAuAZBKD21QM8MF25JZE/sdrxxRFySJtyeJKCrFNqqtNKBan6omh+s5ger+bFaHqdjzuRbkMyGRrAEPX5iADoJQegrUsLZmmvVW8v4/+xyvHpI/Eu66roe+NOmUFYHHa3hh2v44Rp+oIofquFWBw0OZ0MiWJaR3TtAGBSB5aoBPAa/TL5lbAzbPkX95TnxoUJHDx39ZaRqRBTiUOYcnM7U80M1/HANHarmh2v4eTPvG8JSwtmQCLZokDA4nNDgA/AeBKHPYUQ39RKmJAjLT4rT/u3INLIHk4Xx3fF3UCY40dkGfqSGH6mhIzX8SA0/UcdjurFBEWxQON2SyJ4ZLvQPxRU+gK6DIPRRaoHuHSDM6SN8eFJcuN1BRAsGCnP6CiEaqSuDjnC29o7W8mO1dLSGH6vlx2p5VCBLDqdB4eyaOPbQIGFgGPo5AaSEtUa7mtls1us7vOj25lL+/46J60vEW3sLCwYKgyN8vYGozLVGLXY6UcdP1PHjtXSslh+v5SfrefduLDmcksPYgDCWEs4GhrFgP3w3Y7PZHA5HYGCg1IV4HdYaVSB8F/zDhO5sQndVaZNq2QnxN987EoNpXj9hSoIQJf+/S76rqJH/p44OmVTnrY5jtfxEHZVZeN8Q1j+U9Q+jqT3Z4qHCgFDWDb9kAL4NLcKu1rkWYVt2kdZcED89zf9dIg6NZNN6CtN6+twSWTJrEVZY6VQ9/08dP1nHT9bRf+r5yToeoqH+YSwxyD4oSjswjPUPpV7BTK43X0aLUGbQImwL3wX/oxZoei9hei+yOlQbfuJfnRNfPOCIC2LTegnTerIhPt9r6uN+auKn6+lUPT9dz0/V06l6fqqeqxj1CWF9Qlj/MHZTL+obKvQNZc7rtQ0NluBg+ccDgIwhCP1YoIpujGc3xqve5art5fyr8+JN68UGGx8dLYyNYWOi2bBIpsbgw3bUttC5Bn62gZ9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"image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A(n)=3/n+5-4*p^n\n", "plot(A,2:68)" ] }, { "cell_type": "markdown", "id": "f93251a8-aef6-434f-af54-09d8df939229", "metadata": {}, "source": [ "From the graph we see that the minimum of $A(n)$ is somewhere between 10 and 30. Therefore,\n", "we test $A(n)$ for each integer between these values and find the minimum." ] }, { "cell_type": "code", "execution_count": 4, "id": "f02dd58e-f426-47d3-8ed0-34808d34548c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "21×2 Matrix{Float64}:\n", " 10.0 1.37928\n", " 11.0 1.35985\n", " 12.0 1.34495\n", " 13.0 1.33353\n", " 14.0 1.32484\n", " 15.0 1.31833\n", " 16.0 1.3136\n", " 17.0 1.31032\n", " 18.0 1.30824\n", " 19.0 1.30719\n", " 20.0 1.307\n", " 21.0 1.30754\n", " 22.0 1.30872\n", " 23.0 1.31044\n", " 24.0 1.31265\n", " 25.0 1.31527\n", " 26.0 1.31827\n", " 27.0 1.32159\n", " 28.0 1.3252\n", " 29.0 1.32907\n", " 30.0 1.33317" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T=vcat([[n A(n)] for n=10:30]...)" ] }, { "cell_type": "code", "execution_count": 5, "id": "4876480d-5de0-4d7c-bee8-8cdf390c949b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "20" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nmin=Int(T[argmin(T[:,2]),1])" ] }, { "cell_type": "code", "execution_count": 6, "id": "c154c0a0-7a0a-4995-9cc4-0c5fe1543b9a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.3069961718946295" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Amin=A(nmin)" ] }, { "cell_type": "markdown", "id": "2f7a5e38-d0a9-43b0-a268-36bc09c22c59", "metadata": {}, "source": [ "It follows that the minimum occurs at $n=20$ and is $A(20)\\approx 1.3069961718946295$." ] }, { "cell_type": "markdown", "id": "977fa58c-a637-412e-b48a-261441dc1923", "metadata": {}, "source": [ "(iii) Let $q$ be the probability that a diode is faulty.\n", "Compute the relative sensitivity $S(A,q)$ evaluated at $q=0.002$." ] }, { "cell_type": "markdown", "id": "8abd53ac-b6a4-4dfc-8045-2eaee34ce482", "metadata": {}, "source": [ "Assume for infinitesimal small changes the integer $n$ where the minimum occurs doesn't change. Then" ] }, { "cell_type": "code", "execution_count": 7, "id": "f6b58d4e-2d62-4b5d-bd14-679a7d8fcb60", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "D (generic function with 1 method)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Symbolics\n", "D(f,x)=expand_derivatives(Differential(x)(f))" ] }, { "cell_type": "code", "execution_count": 8, "id": "0d656973-8190-4fc5-83de-fcb2861a92b4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2-element Vector{Num}:\n", " p\n", " q" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "@variables p,q" ] }, { "cell_type": "code", "execution_count": 9, "id": "6292a3cf-4547-4adb-9b7b-6d80cbb36708", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[96m5.15\u001b[39m - \u001b[96m4\u001b[39m((\u001b[96m1\u001b[39m - q)^\u001b[96m20\u001b[39m)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Aq=substitute(A(nmin),p=>1-q)" ] }, { "cell_type": "code", "execution_count": 10, "id": "06027382-49e6-46bb-8c63-ca5bfbfc4831", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.11784901352849506" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "DAdq=D(Aq,q)\n", "SAq=substitute(q/Amin*DAdq,q=>0.002)" ] }, { "cell_type": "markdown", "id": "8ee08d9d-8ed3-41af-8ea1-f5a842e0c87b", "metadata": {}, "source": [ "It follows that the relative sensitivity is\n", "$$\n", " S(A,q)={q\\over A}{dA\\over dq}\\Big|_{q=0.002}\\approx 0.11784901352849506.\n", "$$" ] }, { "cell_type": "code", "execution_count": null, "id": "2d4601be-7c23-46e8-8b64-a996ab80e341", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.10", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.10" } }, "nbformat": 4, "nbformat_minor": 5 }