12th Grade

{
"voice_prompt": "Speak clearly and naturally. Pause briefly at commas and for mathematical terms like E of X and mu.",
"manuscript": {
"title": {
"text": "How do you calculate the expected value of a random variable",
"audio": "How do you calculate the expected value of a random variable"
},
"description": {
"text": "You calculate the expected value by multiplying each possible value of a random variable by its probability and then summing the results. The expected value is the average if you repeat the experiment many times.",
"audio": "You calculate the expected value by multiplying each possible value of a random variable by its probability and then summing the results. The expected value is the average if you repeat the experiment many times."
},
"scenes": [
{
"text": "The expected value of a discrete random variable X is written as E of X. It tells you the average value of X if you repeat the experiment many times. But first, what does 'discrete' mean? Discrete means separate values, like whole numbers. For instance zero, one or two. The opposite is continuos, which means values can be any number, even decimals. That's like the whole number line. This video focuses only on discrete random variables.",
"latex": "E(X)"
},
{
"text": "To calculate the expected value, you multiply each possible value of the random variable by its probability, and then you sum up all those products. The formula can be written using sigma notation. ",
"latex": "E(X) = \\sum_{i=1}^{n} x_i P(X=x_i)"
},
{
"text": "Let's look at the die. Where X is the outcome of the roll. The die has 6 sides, and you have equal probability to roll each number. So P of X equals one sixth for every outcome. Now let's calculate the expected value. Remember to calculate the expected value, you multiply every possible value by its probability.",
"latex": "E(X) = x_1 *P(X=1) + x_2 *P(X=2) + x_3 *P(X=3) + x_4 *P(X=4) + x_5 *P(X=5) + x_6 *P(X=6)= "
},
{
"text": "The expected value equals 1 times one sixth plus 2 times one sixth plus 3 times one sixth plus 4 times one sixth plus 5 times one sixth plus 6 times one sixth. This equals 3.5.",
"latex": "E(X) = 1 * \\dfrac{1}{6} + 2 * \\dfrac{1}{6}+ 3 * \\dfrac{1}{6}+ 4 * \\dfrac{1}{6} + 5 * \\dfrac{1}{6} + 6 * \\dfrac{1}{6} = 3.5 "
},
{
"text": "What does an expected value of 3.5 mean? It's the long-run average. It's not necessarily a possible outcome, you can't actually roll 3.5. And it's not the most probable outcome. You can think of it as the 'center of mass' of the probability distribution.",
"latex": ""
}
{
"text": "Why is it useful to know how to calcualte expected values? Let's say you want to buy a ticket for a lottery. There are 100 tickets in total. One of them wins the main price of $1000. In addition, there are 10 smaller prizes where you can win $20. The rest of the tickets win nothing. So, what is the expected value to win in this lottery? Let's find out. The probability of winning the main prize is 1 out of 100. The probability of winning $20 is 10 out of 100, and the probability of winning $0 is 89 over 100. Now you can out this information in the formula.",
"latex": "E(X) = 1000 * \\dfrac{1}{100} + 20 * \\dfrac{10}{100}+ 0 * \\dfrac{89}{100} = 10 + 2 + 0 = 12"
},

{
"text": "The expected value is $12. That means, if you spend more than $12 on each ticket, you should expect to lose money. This gives you a more realistic idea of what you might actually win in this lottert. It helps you make better decisions based on facts - not just hope.",
"latex": ""
},
],
"outro": {
"text": "To calculate the expected value of a random variable you multiplying each possible value by its probability and then summing the results. The expected value is the average if you repeat the experiment many times.",
"audio": "To calculate the expected value of a random variable you multiplying each possible value by its probability and then summing the results. The expected value is the average if you repeat the experiment many times"
}
}
}