MEANING OF 'MEAN-VALUE'

A cricket game in progress. The batting team has scored 60 runs so far in 10 overs. Dividing 60 by 10 we get 6. 6 is the average number of runs per over taken so far. .That is , the team seems to have scored 6 runs in each over (Practically, they might have scored 12 runs in a over and zero runs in another over). 'Average runs' is also called run rate. If the rain bring the game to a halt, run rate is used to decide the winner. Remember, as the game processes, the run rate changes.

Toss a coin 5 times . 4 times you may get the head and 1 time tail may appear. So the mean is 80% for head and 20% for tail. Toss the coin 10 times. 6 times may be head. 4 times may be tail. Now the average is 60% and 40%.
Throw the coin for 1000 times. (A computer simulation helps). You may get the head 502 times and tail 498 times. The average of getting head or tail per throw approaches 50% or 1/2. When the volume of samples or trials go on increasing, the average approaches probability.

The normal values of blood sugar, blood pressure, life expectancy, normal height, normal rainfall, per-capita income,all are averages based on many samples and studies.

Take 10 persons. Measure their heights. Find their average height. Find at least 30 samples of 10 persons and calculate their average height(30 averages). Now draw a graph. Taking averages in X- axis and heights in Y- axis. A bell curve or normal distribution curve appears. The curve tells us that majority of averages are close to the universal average. Averages peak or accumulate at the center. That is the reason for bell shape. This concept is termed as 'central limit theorem'. This is applicable to all the natural attributes like height, weight, blood sugar, intelligence, courage and so on. So we may be average in many traits but best in a few traits.

Moving averages are used in stock market. Average price of a stock is calculated for a week. When the new week day is born, most old price value is shed from the list, the new day's price value is taken into account and fresh average price is calculated. The average price runs(changes) everyday. Investors can speculate using the running average and can decide to buy, sell or keep it.

The mean value is the safest value. When it is difficult to take a firm a decision, take shelter under average value.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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