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You get to use established statistical formulas for finding binomial probabilities, using the values of n and p unique to each problemThe Probability Mass Function (PMF) also called a probability function or frequency function which characterizes the distribution of a discrete random variable Let X be a discrete random variable of a function, then the probability mass function of a random variable X is given by P x (x) = P( X=x ), For all x belongs to the range of X It is noted that the probability function should fall onTo overcome this issue we can calculate an annualized ROI formula ROI Formula = (Ending Value / Beginning Value) ^ (1 / # of Years) – 1 Where

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P(x) formula example-P(x = 1) = 4!1!3!F(a) = p(X a) = p(x) (for discrete variables) X a ∑ ≤ ≤ G EXAMPLE – DISCRETE CASE Probability calculations are often very simple when one is dealing with a discrete random variable where only a very few values can occur See Hayes, pp 9596, for an example of an experiment involving rolling two dice Here is another example

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For example, P (x) = 4x 2 2x – 9In common usage, they are sometimes just called "polynomials" For realvalued polynomials, the general form is p (x) = p n x n p n1 x n1 p 1 x p 0 The univariate polynomial is called a monic polynomial if p n ≠ 0 and it is normalized to p n = 1 (Parillo, 06)P = Percentile Formula for Percentile In mathematics, we use this term percentile informally to indicate that a certain percentage falls below that percentile value For example, if we score in the 25th percentile, then 25% of testtakers are below this score Here 25 is called the percentile rankGiven that X is a Poisson random variable with {eq}\lambda {/eq} = 05, use the formula to determine the following probabilities a P(X = 0)

For example, a return of 25% over 5 years is expressed the same as a return of 25% over 5 days But obviously, a return of 25% in 5 days is much better than 5 years!P(X= 129) = P(X= 129 and Y = 15) P(X= 129 and Y = 16) = 012 008 = 0 What is the probability distribution of X?Marginal Probability Mass Function If X and Y are discrete random variables with joint probability mass function fXY(x;y), then the marginal probability mass functions of Xand Y are fX(x) = X y fXY(x;y) and fY(y) = X x fXY(x;y) where the sum for fX(x) is over all points in the range of (X;Y) for which X= xand the sum for fY(y) is over all

P (X = 1) = 1/6, P (X = 2) = 1/6, etc So E (X 2) = 1/6 4/6 9/6 16/6 25/6 36/6 = 91/6 = The expected value of a constant is just the constant, so for example E (1) = 1 Multiplying a random variable by a constant multiplies the expected value by that constant, so E 2X = 2E XThe rational function f(x) = P(x) / Q(x) in lowest terms has an oblique asymptote if the degree of the numerator, P(x), is exactly one greater than the degree of the denominator, Q(x) You can find oblique asymptotes using polynomial division, where the quotient is the equation of the oblique asymptoteP{X ≥ Y} = Z 1 0 x 0 4 5 (xy xy)dydx = 4 5 1 0 1 2 xy2 xy 1 2 y2 x 0 dx = 4 5 Z 1 0 1 2 x3 3 2 x2 dx = 4 5 1 8 x4 1 2 x3 1 0 = 4 5 · 5 8 = 1 2 The joint cumulative distribution function is defined as F X,Y (x,y) = P{X ≤ x,Y ≤ y} For the case of continuous random variables, we have F X,Y (x,y) = Z y −∞ x −∞ f X,Y (s,t)dtds 3

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Cost, Revenue & Profit Examples 1) A softdrink manufacturer can produce 1000 cases of soda in a week at a total cost of $6000, Since this is a linear function, we can find using the formula for finding the slope of a line between two points, where the points here are and The slope formula isThe Binomial Formula Explained Each piece of the formula carries specific information and completes part of the job of computing the probability of x successes in n independ only2event (success or failure) trials where p is the probability of success on a trial and q is the probability of failure on the trialA polynomial function is a function that involves only nonnegative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etcFor example, 2x5 is a polynomial that has exponent equal to 1 Study Mathematics at BYJU'S in a simpler and exciting way here A polynomial function, in general, is also stated as a polynomial or

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Cost, Revenue & Profit Examples 1) A softdrink manufacturer can produce 1000 cases of soda in a week at a total cost of $6000, the cost C as a function of x, the number of tuxedos rented Solution Since it costs $ regardless of how many tuxes you rent, this is the fixed cost Since∑ xϵRange(x) P x (x) = 1 Here the Range(X) is a countable set and it can be written as { x 1 , x 2 , x 3 , } This means that the random variable X takes the value x 1 , x 2 , x 3 ,Example 32 If X 1 and X 2 are independent binomial random variables with respective parameters (n 1, p) and (n 2, p), calculate the conditional probability mass function of X 1 given that X 1 X 2 = m Solution

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Univariate Polynomial A univariate polynomial has one variable—usually x or tFor example, P(x) = 4x 2 2x – 9In common usage, they are sometimes just called "polynomials" For realvalued polynomials, the general form is p(x) = p n x n p n1 x n1 p 1 x p 0 The univariate polynomial is called a monic polynomial if p n ≠ 0 and it is normalized to p n = 1 (ParilloHere is the law's formula and an example of how to use it Boyle's law says the volume of a mass of gas is inversely proportional to its pressure Here is the law's formula and an example of how to use it Menu Home Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature HumanitiesExample 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the first

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If x is used as the name of an object inside the function, the value of x in the global environment (outside the function) does not change x < 1 fun2 < function() { x < 2 print(x) } fun2() # 2 x #1 To change the global value of a variable inside a function you can use the double assignment operator (Given random variables,, , that are defined on a probability space, the joint probability distribution for ,, is a probability distribution that gives the probability that each of ,, falls in any particular range or discrete set of values specified for that variable In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to anyP(Event) = (Number of ways event can occur) * P(One occurrence) Suppose, for example, we want to find the probability of getting 4 heads in 10 tosses In this case, we'll call getting a heads a "success" Also, in this case, n = 10, the number of successes is r = 4, and the number of failures (tails) is n – r = 10 – 4 = 6

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Even though the function my_func() changed the value of x to 10, it did not affect the value outside the function This is because the variable x inside the function is different (local to the function) from the one outsideTherefore, our profit function equation will be as follows P(x) = R(x) C(x) Examples Let's show a simple example A street vendor in New York City sells hot dogs for $300 eachLet us assume that the quantity demanded of a commodity X is D x, which depends only on its price P x, while other factors are constantIt can be mathematically represented as D x = f (P x) However, the quantitative relationship between D x and P x is expressed as D x = a – bP x Where, a (intercept) and b (relationship between D x and P x) are constants Also Read Law of Demand

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The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring The expected value of X is usually written as E(X) or m E(X) = S x P(X = x) So the expected value is the sum of (each of the possible outcomes) × (the probability of theP(x) is the probability of x successes occur in the n number of events, p is the probability of success and q is the probability of failure often denoted by q = (1 p) The binomial distribution arise for the following 4 conditions, when the event hasThat just means "the probability that X takes on some value x" It's not a very useful equation on its own;

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The STDEVP Function is categorized under Excel Statistical functions STDEVP will calculate the standard deviation that is based on an entire population given as arguments It will ignore logical values and text As a financial analyst, the stdevp function can be useful in calculating deviations in revenueSupport set together with the halfplane y ≤ −xz, we get FZ(z) = P(Z ≤ z) = P(X Y ≤ z) = P(Y ≤ −X z) = ˆ0 if z < 0 Rz 0 Rz−x 0 λe −λxλe−λydydx = 1−e−λz −λze−λz if z ≥ 0 This gives the pdf, fZ(z) = ˆ 0 if z < 0 λ2ze−λz if z ≥ 0, which is the pdf of a Gamma(2,λ) Thus, Z is Gamma(2,λ) random variable (b)The probability mass function, P ( X = x) = f ( x), of a discrete random variable X is a function that satisfies the following properties P ( X = x) = f ( x) > 0, if x ∈ the support S ∑ x ∈ S f ( x) = 1 P ( X ∈ A) = ∑ x ∈ A f ( x) First item basically says that, for every element x in the support S, all of the probabilities must be positive

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The function f(x) = 2x3 Put "y" for "f(x)" y = 2x3 Subtract 3 from both sides y3 = 2x Divide both sides by 2 (y3)/2 = x Swap sides x = (y3)/2 Solution (put "f1 (y)" for "x") f1 (y) = (y3)/2Conditioning on the discrete level Example A fair coin is tossed 10 times;L' (p ) = Σ x i p1 Σ x i (1 p) n Σ x i (n Σ x i)p Σ x i (1 p) n1 Σ x i We rewrite some of the negative exponents and have L' ( p ) = (1/ p ) Σ x i p Σ x i (1 p ) n Σ x i 1/(1 p ) ( n Σ x i )p Σ x i (1 p ) n Σ x i

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PValue in Excel – Example #2 Here for let's assume some values to determine the support against qualifying the evidence For our formula =TDIST (x, deg_freedom, tails) Here if we take x=t (test statistics), deg_freedom = n, tail = 1 or 2 Here as we can see the results, if we can see in percentages it's 272%× (1/6) 2 (5/6) 2 = 6 × (1/6) 2 × (5/6) 2 = P(X = 3) = 4!What's more useful is an equation that tells you the probability of some individual event happening For example P(X = 1) = 02 * 02 How you come up with these equations depends mostly on what type of event you have

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The random variable X is the number of heads in these 10 tosses, and Y — the number of heads in the first 3 tosses In spite of the fact that Y emerges before X it may happen that someone knows X but not Y Conditional probability× (1/6) 3 (5/6) 1 = 4 × (1/6) 3 × (5/6) =F(x) = P(X ≤ x) Continuous distribution The cumulative distribution function F(x) is calculated by integration of the probability density function f(u) of continuous random variable X Discrete distribution The cumulative distribution function F(x) is calculated by summation of the probability mass function P(u) of discrete random variable X

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End Behavior of a Function The end behavior of a polynomial function is the behavior of the graph of f (x) as x approaches positive infinity or negative infinity The degree and the leading coefficient of a polynomial function determine the end behavior of the graph The leading coefficient is significant compared to the other coefficients in the function for the very large or very small\(\sigma _{\bar{x}}=\frac{3217 }{\sqrt{40}}\) = Now, using the test static formula, we get t = ( – 1) / Therefore, t = Using the ZScore table, we can find the value of P(t>) From the table, we get P (t) = 0003 Therefore, If P(t>) =1 0003 =0997 P value =0997 > 005Simply as P(x) Likewise, P(X ≤ x) = probability that the random variable X is less than or equal to the specific value x;

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After you identify that a random variable X has a binomial distribution, you'll likely want to find probabilities for X The good news is that you don't have to find them from scratch;The Binomial Formula Explained Each piece of the formula carries specific information and completes part of the job of computing the probability of x successes in n independ only2event (success or failure) trials where p is the probability of success on a trial and q is the probability of failure on the trialBasic Excel Formulas Guide Mastering the basic Excel formulas is critical for beginners to become highly proficient in financial analysis Financial Analyst Job Description The financial analyst job description below gives a typical example of all the skills, education, and experience required to be hired for an analyst job at a bank, institution, or corporation

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The function pX(x)= P(X=x) for each x within the range of X is called the probability distribution of X It is often called the probability massfunction for the discrete random variable X Example 1 Find a formula for the probability distribution of the total number of heads obtained in four tossesof a balanced coin The samplespaceFormulas — you just can't get away from them when you're studying statistics Here are ten statistical formulas you'll use frequently and the steps for calculating them Proportion Some variables are categorical and identify which category or group an individual belongs to For example, "relationship status" is a categorical variable, and an individual could beP(X ≥xyX ≥x) = P(X ≥xy,X ≥x) P(X ≥x) = P(X ≥xy) P(X ≥x) = e−λ(xy) e−λx = e−λy = P(X ≥y), which proves the memoryless property The memoryless property also has a geometric interpretation Consider the probability density function of an exponential(λ) random variable from some value x to infinity The area under the

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PX(x)=P(X = x)= 2 x 10 3−x 12 3 forx=0, 1, 2 (13) For example p(0) = P(X =0)= 2 0 10 3−0 12 3 = (1)(1) 2 = 6 11 (14) We collect this informationas in table 4 TABLE 4 Probabilities for Television Problem x 0 1 2 pX(x) 6/11 9/22 1/22 FX(x) 6/11 21/22 1 We computethe expected value as E(X)= X x X xpX (x) = (0) 6 11 (1) 9 22 (2) 1 22 = 11 22 = 1 2 (15) 18E(x) = ∑x * P(x) The formula, by the way, shows the same thing you have seen in the examples before it's the weighted mean of the possible outcomes, where the weight is the probability of each event occurring Examples of applying and calculating Expected Value Let me give you a few more reallife examples to hammer home the concept and× (1/6) 1 (5/6) 3 = 4 × (1/6) × (5/6) 3 = P(X = 2) = 4!

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P(a ≤ X ≤ b) = probability that X lies between values a and b Harnett, on the other hand, likes to use boldface italic for rvs, and hence in his notation P(x = x) symbolizesP Value = Note Excel directly gives the pvalue using the formula CHITEST (actual range, expected range) Example #4 It is known that 60% of the people who enter apparel stores in a city purchase something An apparel shop owner wanted to find if the number is higher for the apparel shop owned by him

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