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Roger Labbe 2015-12-13 15:16:33 -08:00
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03-Gaussians.ipynb
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@ -294,7 +294,7 @@
"### Random Variables\n",
"\n",
"\n",
"If we roll a die we can get a value between 1 and 6. Each possible value has a probability of occuring attached to it: 1/6 for a fair die. This combination of possible values within some range and associated probabilities is called a *random variable*, and the range of values is called the *sample space* These names are based on the idea that we will be randomly sampling from a population. If we rolled a fair die many times we'd expect the value '4' to occur 1/6 of the time. 'Random' does not mean the process is random - rolling a die is entirely deterministic - only that the sampling is random.\n",
"If we roll a die we can get a value between 1 and 6. Each possible value has a probability of occuring attached to it: 1/6 for a fair die. This combination of values and associated probabilities is called a *random variable*, and the range of values is called the *sample space* These names are based on the idea that we will be randomly sampling from a population. If we rolled a fair die many times we'd expect the value 4 to occur 1/6 of the time. 'Random' does not mean the process is random - rolling a die is entirely deterministic. However, prior to the roll we cannot predict the outcome except in a probablistic manner.\n",
"\n",
"Another example of a random variable is the heights of students in a university. Here the sample space is a range of values in the real numbers between two limits defined by biology. This is random in the sense that we can randomly select students from random classes; the height of a student is determined by genetics, nutrition, age, and other factors.\n",
"\n",
@ -327,14 +327,14 @@
"\n",
"$$P(X{=}4) = f(4) = \\frac{1}{6}$$\n",
"\n",
"This states that the probability of the die landing on 4 is $\\frac{1}{6}$. $P(X{=}x_k)$ is notation for \"the probability of $X$ being $x_k$. Some texts use $Pr$ or even $Prob$ instead of $P$.\n",
"This states that the probability of the die landing on 4 is $\\frac{1}{6}$. $P(X{=}x_k)$ is notation for \"the probability of $X$ being $x_k$. Some texts use $Pr$ or $Prob$ instead of $P$.\n",
"\n",
"\n",
"Another example is a fair coin. It has the sample space {H, T}. The coin is fair, so the probability for heads (H) is 50%, and the probability for tails (T) is 50%. We write this as\n",
"\n",
"$$P(X{=}H) = 0.5\\\\P(X{=}T)=0.5$$\n",
"\n",
"As an aside, sample spaces are not unique. One sample space for a die is {1, 2, 3, 4, 5, 6}. Another valid sample space would be {even, odd}. Another might be {dot in center, no dot in center}. A sample space is valid so long as it covers all possibilities, and any single event is described by only one element. {even, 1, 3, 4, 5} is not a valid sample space for a die since a value of 4 is matched both by 'even' and '4'.\n",
"Sample spaces are not unique. One sample space for a die is {1, 2, 3, 4, 5, 6}. Another valid sample space would be {even, odd}. Another might be {dots in all corners, not dots in all corners}. A sample space is valid so long as it covers all possibilities, and any single event is described by only one element. {even, 1, 3, 4, 5} is not a valid sample space for a die since a value of 4 is matched both by 'even' and '4'.\n",
"\n",
"The probabilities for all values of a *discrete random value* is known as the *discrete probability distribution* and the probabilities for all values of a *continuous random value* is known as the *continuous probability distribution*.\n",
"\n",