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3 years ago

A runner competes in a local marathon. He places 34th which is exactly the top 40% of all runners. How many runners competed

Mathematics

1 answer:

laiz [17]3 years ago

6 0

Answer:

85 runners

Step-by-step explanation:

we know that

34th represent 40% of all runners

using proportion

Find out the total number of runners

$\frac{34}{40\%}=\frac{x}{100\%}\\\\x=34(100)/40\\\\x= 85\ runners$

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Let X and Y be discrete random variables. Let E[X] and var[X] be the expected value and variance, respectively, of a random vari

Ulleksa [173]

Answer:

(a) $E[X+Y]=E[X]+E[Y]$

(b) $Var(X+Y)=Var(X)+Var(Y)$

Step-by-step explanation:

Let X and Y be discrete random variables and E(X) and Var(X) are the Expected Values and Variance of X respectively.

(a)We want to show that E[X + Y ] = E[X] + E[Y ].

When we have two random variables instead of one, we consider their joint distribution function.

For a function f(X,Y) of discrete variables X and Y, we can define

$E[f(X,Y)]=\sum_{x,y}f(x,y)\cdot P(X=x, Y=y).$

Since f(X,Y)=X+Y

$E[X+Y]=\sum_{x,y}(x+y)P(X=x,Y=y)\\=\sum_{x,y}xP(X=x,Y=y)+\sum_{x,y}yP(X=x,Y=y).$

Let us look at the first of these sums.

$\sum_{x,y}xP(X=x,Y=y)\\=\sum_{x}x\sum_{y}P(X=x,Y=y)\\\text{Taking Marginal distribution of x}\\=\sum_{x}xP(X=x)=E[X].$

Similarly,

$\sum_{x,y}yP(X=x,Y=y)\\=\sum_{y}y\sum_{x}P(X=x,Y=y)\\\text{Taking Marginal distribution of y}\\=\sum_{y}yP(Y=y)=E[Y].$

Combining these two gives the formula:

$\sum_{x,y}xP(X=x,Y=y)+\sum_{x,y}yP(X=x,Y=y) =E(X)+E(Y)$

Therefore:

$E[X+Y]=E[X]+E[Y] \text{ as required.}$

(b)We want to show that if X and Y are independent random variables, then:

$Var(X+Y)=Var(X)+Var(Y)$

By definition of Variance, we have that:

$Var(X+Y)=E(X+Y-E[X+Y]^2)$

$=E[(X-\mu_X +Y- \mu_Y)^2]\\=E[(X-\mu_X)^2 +(Y- \mu_Y)^2+2(X-\mu_X)(Y- \mu_Y)]\\$Since we have shown that expectation is linear$\\=E(X-\mu_X)^2 +E(Y- \mu_Y)^2+2E(X-\mu_X)(Y- \mu_Y)]\\=E[(X-E(X)]^2 +E[Y- E(Y)]^2+2Cov (X,Y)$

Since X and Y are independent, Cov(X,Y)=0

$=Var(X)+Var(Y)$

Therefore as required:

$Var(X+Y)=Var(X)+Var(Y)$

7 0

2 years ago

HELP!

maksim [4K]

Part A

$4x^2 -7x-15=0\\\\(4x+5)(x-3)=0\\\\x=-\frac{5}{4}, 3$

So, the x-intercepts are $\boxed{\left(-\frac{5}{4}, 0 \right), (3,0)}$

Part B

The vertex will be a minimum because the coefficient of $x^2$ is positive.

The x-coordinate of the vertex is $x=-\frac{-7}{2(4)}=\frac{7}{8}$

Substituting this back into the function, we get $f\left(\frac{7}{8} \right)=4\left(\frac{7}{8} \right)^2 -7\left(\frac{7}{8} \right)^2 -15=-\frac{289}{16}$

So, the coordinates of the vertex are $\boxed{\left(\frac{7}{8}, -\frac{289}{16} \right)}$

Part C

Plot the vertex and the x-intercepts and draw a parabola that passes through these three points.

The graph is shown in the attached image.

7 0

1 year ago