10.4: Discrete Random Variables
Let \((S,P)\) be a probability space and let \(X:S \rightarrow \mathbb{R}\) be any function that maps the outcomes in \(S\) to real numbers (all values allowed, positive, negative and zero). We call \(X\) a random variable . The quantity \(\sum_{x \in S} X(x)P(x)\), denoted \(E(X)\), is called the expectation (also called the mean or expected value ) of the random variable \(X\). As the suggestive name reflects, this is what one should expect to be the average behavior of the result of repeated Bernoulli trials.
Note that since we are dealing only with probability spaces \((S,P)\) where \(S\) is a finite set, the range of the probability measure \(P\) is actually a finite set. Accordingly, we can rewrite the formula for \(E(X)\) as \(\sum_{y} y \cdot [rpb(X(x) = y)\), where the summation extends over a finite range of values for \(y\).
For the spinner shown in Figure 10.1 , let \(X(i)=i^2\) where \(i\) is the number of the region. Then
\(E(x) = \displaystyle \sum_{i \in S} i^2P(i) = 1^2 \dfrac{1}{8} + 2^2 \dfrac{2}{8} + 3^2 \dfrac{1}{8} + 4^2 \dfrac{1}{8} + 5^2 \dfrac{3}{8} = \dfrac{109}{8}\).
Note that \(109/8=13.625\). The significance of this quantity is captured in the following statement. If we record the result from the spinner \(n\) times in succession as \((i_1,i_2,…,i_n)\) and Xing receives a prize worth \(i_j^2\) for each \(j=1,2,…,n\)), then Xing should “expect” to receive a total prize worth \(109n/8=13.625n\). Bob asks how this statement can possibly be correct, since \(13.625n\) may not even be an integer, and any prize Xing receives will have integral value. Carlos goes on to explain that the concept of expected value provides a formal definition for what is meant by a fair game. If Xing pays 13.625 cents to play the game and is then paid \(i^2\) pennies where i is the number of the region where the spinner stops, then the game is fair. If he pays less, he has an unfair advantage, and if he pays more, the game is biased against him. Bob says “How can Xing pay 13.625 pennies?” Brushing aside Bob's question, Carlos says that one can prove that for every \(ϵ>0\), there is some \(n_0\) (which depends on \(ϵ\)) so that if \(n>n_0\), then the probability that Xing's total winnings minus \(13.625n\), divided by \(n\) is within \(ϵ\) of 13.625 is at least \(1−ϵ\). Carlos turns to Dave and explains politely that this statement gives a precise meaning of what is meant by “close” and “large”.
For Alice's game from the start of the chapter, \(S=\{0,1,2,3,4,5\}\), we could take \(X\) to be the function defined by \(X(d)=2−d\). Then \(X(d)\) records the amount that Bob wins when the difference is \(d\) (a negative win for Bob is just a win for Alice in the same amount). We calculate the expectation of \(X\) as follows:
\(E(X) = \displaystyle \sum_{d=0}^5 X(d)p(d) = -2 \dfrac{1}{6} - 1 \dfrac{10}{36} + 0 \dfrac{8}{36} + 1 \dfrac{6}{36} + 2 \dfrac{4}{36} + 3236 = \dfrac{-2}{36}\).
Note that \(−2/36=−.055555….\) So if points were dollars, each time the game is played, Bob should expect to lose slightly more than a nickel. Needless to say, Alice likes to play this game and the more times Bob can be tricked into playing, the more she likes it. On the other hand, by this time in the chapter, Bob should be getting the message and telling Alice to go suck a lemon.
10.4.1 The Linearity of Expectation
The following fundamental property of expectation is an immediate consequence of the definition, but we state it formally because it is so important to discussions to follow.
Let \((S,P)\) be a probability space and let \(X_1,X_2,…,X_n\) be random variables. Then
\(E(X_1 + X_2 + \cdot \cdot \cdot + X_t) = E(X_1) + E(X_2) + \cdot \cdot \cdot + E(X_n)\).
10.4.2 Implications for Bernoulli Trials
Consider a series of \(n\) Bernoulli trials with \(p\), the probability of success, and let \(X\) count the number of successes. Then, we claim that
\(E(X) = \displaystyle \sum_{i=0}^n i \dbinom{n}{i} p^i(1-p)^{n-i} = np\)
To see this, consider the function \(f(x)=[px+(1−p)]^n\). Taking the derivative by the chain rule, we find that \(f′(x)=np[px+(1−p)]^{n−1}\). Now when \(x=1\), the derivative has value \(np\).
On the other hand, we can use the binomial theorem to expand the function \(f\).
\(f(x) = \displaystyle \sum_{i=0}^n \dbinom{n}{i} x^ip^i(1-p)^{n-i}\)
It follows that
\(f'(x) = \displaystyle \sum_{i=0}^n i \dbinom{n}{i} x^{i-}p^i(1-p)^{n-i}\)
And now the claim follows by again setting \(x=1\). Who says calculus isn't useful!
Many states have lotteries to finance college scholarships or other public enterprises judged to have value to the public at large. Although far from a scientific investigation, it seems on the basis of our investigation that many of the games have an expected value of approximately fifty cents when one dollar is invested. So the games are far from fair, and no one should play them unless they have an intrinsic desire to support the various causes for which the lottery profits are targeted.
By contrast, various games of chance played in gambling centers have an expected return of slightly less than ninety cents for every dollar wagered. In this setting, we can only say that one has to place a dollar value on the enjoyment derived from the casino environment. From a mathematical standpoint, you are going to lose. That's how they get the money to build those exotic buildings.


7.4: Brain Maturation
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The brain is about 75 percent its adult weight by two years of age. By age 6, it is approximately 95 percent its adult weight. Myelination and the development of dendrites continues to occur in the cortex and as it does, we see a corresponding change in the child’s abilities. Significant development in the prefrontal cortex (the area of the brain behind the forehead that helps us to think, strategize, and control emotion) makes it increasingly possible to control emotional outbursts and to understand how to play games. Consider 4- or 5-year-old children and how they might approach a game of soccer. Chances are, every move would be a response to the commands of a coach standing nearby calling out, “Run this way! Now, stop. Look at the ball. Kick the ball!” And when the child is not being told what to do, he or she is likely to be looking at the clover on the ground or a dog on the other side of the fence! Understanding the game, thinking ahead, coordinating movement, and handling losing improve with practice and myelination.7
Children’s drawings are representative of the development of visual pathways; as children’s brains mature the images in their drawings change. Early scribbles and dots illustrate the use of simple motor skills. No real connection is made between an image being visualized and what is created on paper.
At age 3, the child begins to draw wispy creatures with heads and not much other detail. Gradually pictures begin to have more detail and incorporate more parts of the body. Arm buds become arms and faces take on noses, lips and eventually eyelashes.
Between ages 3 and 6, the left hemisphere of the brain grows dramatically. This side of the brain or hemisphere is typically involved in language skills. The right hemisphere continues to grow throughout early childhood and is involved in tasks that require spatial skills such as recognizing shapes and patterns. The corpus callosum which connects the two hemispheres of the brain undergoes a growth spurt between ages 3 and 6 and results in improved coordination between right and left hemisphere tasks.
7. Lifespan Development - Module 5: Early Childhood by Lumen Learning references Psyc 200 Lifespan Psychology by Laura Overstreet, licensed under CC BY 4.0


8.4.3: Safe Food Handling
When handling food, it is important to keep safety at the forefront of food preparation. You can read more about safe food handling guidelines developed by the United States Department of Agriculture at: http :// www . fsis . usda . gov / wps / wcm / connect / 18 cece 94 - 747 b - 44 ca - 874 f - 32 d 69 fff 1 f 7 d / Basics _ for _ Safe _ Food _ Handling . pdf ? MOD = AJPERES # page =2
Home Health Aides / Personal Care Aides should follow these guidelines for safe food handling :
- ALWAYS wash your hands before and after food preparation. Follow proper hand washing guidelines. Always wear gloves when handling food.
- ALWAYS wash your hands after handling raw meat, poultry, fish, or eggs.
- Use one cutting board for raw meats, poultry, and fish. Use another one for vegetables, fruits, and foods that are ready to eat, such as bread.
- Wash cutting boards in hot soapy water or run them through the dishwasher.
- Non Porous acrylic, plastic, glass, or wood cutting boards can be safely washed in the dishwasher. Discard cutting boards that are worn or that have a lot of grooves where bacteria can hide and contaminate food.
- Immediately wipe up spills from raw meat, poultry, and fish using an antibacterial cleanser or a diluted bleach solution.
- Clean the workspace before and after food preparation using hot soapy water. Clean the workspace and wash your hands after handling raw meats, poultry, or fish before moving on to the next step in food preparation.
- Cutting boards and work surfaces can be sanitized using an antibacterial cleanser or a homemade solution of 1 tablespoon liquid chlorine bleach to 1 gallon of water. Spray solution onto surface, allow a few minutes to work, rinse, and then let air dry or pat dry with a paper towel.
- Keep raw meat, poultry, and fish separate from other items in your food cart to avoid juices contaminating other food. Place these items in plastic bags separate from other food.
- When storing raw meat, poultry, and fish in the refrigerator, place them in containers or on a dish to prevent juices from dripping on other foods. Store these items on the lowest shelf in the refrigerator to prevent dripping from higher shelves onto food stored on lower shelves.
- Your clothes and apron should always be clean.
- Avoid coughing or sneezing during food preparation. If you cough or sneeze, step away from the food preparation area and wash your hands immediately afterwards.
- Use clean dishes, bowls, pots, and dish towels.
- Use hot, soapy water to wash dishes.
- Never taste food and stir food with the same spoon. If you need to taste food, or the patient would like to taste the food during food preparation, use a clean spoon to remove some from the pot or pan, and then discard that spoon into the sink to be washed.
- Hot foods can be placed in the refrigerator immediately. Do not let them sit on the stove or counter for several hours. This encourages bacteria to multiply.
- NEVER use a damaged can or a can that is bulging or dented. It could be spoiled and you risk food poisoning.
- Do not eat raw eggs or use eggs that have cracks in them. Never undercook eggs. You risk salmonella if you do so. Store eggs in their original container in the refrigerator.
- Use a food thermometer to ensure that meat is cooked to the proper doneness.
- When using a microwave, rotate the dish while cooking to avoid cold spots where bacteria may live and multiply.
- When reheating food, bring it up to proper temperature. Sauces, gravies, and soups should be brought to a rolling boil before eating.
- NEVER use foods that are moldy or have a bad smell to them.
- Keep hot foods hot (above 140 degrees F).
- Keep cold foods cold (at or below 40 degrees F).
- Do not refreeze food that has already been frozen.
- Always use food within the recommended time and by the expiration date. When in doubt, do not use it and ask a supervisor for guidance.
| Safe Temperature | |
|---|---|
| Hot Foods | Keep at 140 degrees F or above |
| Cold Foods | Keep at 40 degrees F or below |
Source: United States Department of Agriculture (2013a)
True or False
1. You should wash your hands, cutting boards, and work surfaces immediately after handling raw meat, poultry, and fish. True or False? _________
2. Separate cutting boards should be used for raw meats and fresh vegetables. True or False? _________
3. Never eat or serve raw eggs or eggs that have cracks in them. True or False? _________
- Answer
-
1. True
2. True
3. True
FEEDBACK :
1. To avoid the risk of contamination from raw meat, poultry, and fish, hands, cutting boards and work surfaces should be washed with hot soapy water after use.
2. To avoid the risk of contamination from raw meats, poultry, and fish, use separate cutting boards for meats and fruits and vegetables.
3. Raw eggs and those with cracks in their shell should not be used. They should be discarded as the risk of salmonella is high.