Probability Questions and others ?

[ngkamsengpeter]

Please help me to solve the following questions :
1. Find the equations of both the straight lines that are nclined at an angle of 45 degree with straight line 2x+y-3=0 and passing through point (-1,4)

2. S denotes the sum of first n terms of a geometric series 3 - 1 + 1/3 -... and I denotes the sum of infinity of this series.Find the smallest n such that |I-S|<0.0001

3. The number of accidents that occur on an assembly line has a Poisson distribution with a mean of 3 accidents a year . Find the probability that one randomly chosen year is free from accident and the probability that 2 years out of 5 years chosen at random are free from accidents .

4 .Continuous random variable X has the following probability density function:
f(x)={ 1/8 0<x<9;
0 otherwise
Continuous random variable Y is defined by Y=In X .show that P(Y<y)=1/8
Thanks .:!)

 

[HallsofIvy]

You are going to have to show some work of your own. People aren't just going to give you the answers. To help you we have to know where you are having problems. Also there may be several different methods of solving equations appropriate to different levels.

Please help me to solve the following questions :
1. Find the equations of both the straight lines that are nclined at an angle of 45 degree with straight line 2x+y-3=0 and passing through point (-1,4)

Do you know how to find the angle the line 2x+ y- 3= 0 makes with the x-axis?

2. S denotes the sum of first n terms of a geometric series 3 - 1 + 1/3 -... and I denotes the sum of infinity of this series.Find the smallest n such that |I-S|<0.0001

Do you know the formula for the finite and infinite sums of a geometric series? In particular, what is I?

3. The number of accidents that occur on an assembly line has a Poisson distribution with a mean of 3 accidents a year . Find the probability that one randomly chosen year is free from accident and the probability that 2 years out of 5 years chosen at random are free from accidents .

Okay, what is the formula for probability of a Poisson distribution?

4 .Continuous random variable X has the following probability density function:
f(x)={ 1/8 0<x<9;
0 otherwise
Continuous random variable Y is defined by Y=In X .show that P(Y<y)=1/8
Thanks .:!)

First, I have no idea what you mean by "In X". Could you mean the natural logarithm, ln(X)? Second, f(x) can't possibly be what you give. If it were, the total probability of x being anything would be 9/8 which is impossible- it's larger than 1. Do you mean "f(x)= 1/8 0< x< 8, 0 otherwise"? Finally, I don't believe "P(Y< y)= 1/8" is correct. P(Y< y) will be a function of y. Is it possible you meant P(Y< 0)= 1/8?

 

[ngkamsengpeter]

You are going to have to show some work of your own. People aren't just going to give you the answers. To help you we have to know where you are having problems. Also there may be several different methods of solving equations appropriate to different levels.


Do you know how to find the angle the line 2x+ y- 3= 0 makes with the x-axis?


Do you know the formula for the finite and infinite sums of a geometric series? In particular, what is I?


Okay, what is the formula for probability of a Poisson distribution?


First, I have no idea what you mean by "In X". Could you mean the natural logarithm, ln(X)? Second, f(x) can't possibly be what you give. If it were, the total probability of x being anything would be 9/8 which is impossible- it's larger than 1. Do you mean "f(x)= 1/8 0< x< 8, 0 otherwise"? Finally, I don't believe "P(Y< y)= 1/8" is correct. P(Y< y) will be a function of y. Is it possible you meant P(Y< 0)= 1/8?

I have no idea howto start with questions 1 .

For question 2 , i do know the formulae but when I do it , it become very complicated . The I is the infinite sums of a geometric series .

For question 3 , I can do the first section but I don't know how to do the second section.
For question 4 , is is the natural logarithm, ln(X) , f(x)= 1/8 0< x< 8, and
P(Y< y)= 1/8 (e^y -1) .I also cannot solve this one .

So please help me .

 

[HallsofIvy]

1. Find the equations of both the straight lines that are nclined at an angle of 45 degree with straight line 2x+y-3=0 and passing through point (-1,4)

The slope of the graph of the equation y= mx+ b is m and that is equal to the tangent of the angle the line makes with the x-axis: $tan(\theta)= m$. What is the slope for your line and what is the angle?
Also
$$tan(\theta+ \phi)= \frac{tan(\theta)+ tan(\phi)}{1- tan(\theta)tan(\phi)}$$
In particular, tan(45 degrees)= 1 so
$$tan(\theta+ 45)= \frac{1+ tan(\theta)}{1- tan(\theta)}$$
while
$$tan(\theta- 45)= \frac{-1+ tan(\theta)}{1+ tan(\theta)}$$

2. S denotes the sum of first n terms of a geometric series 3 - 1 + 1/3 -... and I denotes the sum of infinity of this series.Find the smallest n such that |I-S|<0.0001

For the geometric series a+ ar+ ar²+ ...+ arⁿ the sum to the n term is
$$S= a\frac{1- r^{n+1}}{1- r}$$
and the infinite sum is
$$I= a\frac{1}{1- r}$$
How complicated could that be? What are a and r for your example and what is I?

3. The number of accidents that occur on an assembly line has a Poisson distribution with a mean of 3 accidents a year . Find the probability that one randomly chosen year is free from accident and the probability that 2 years out of 5 years chosen at random are free from accidents .

I'm glad to hear that you can do the first part, finding the probability that there are no accidents in a given year- I'll call that p. To find the probability that there are no accident in 2 of 5 given years, use the binomial distribution with probabilities p and 1- p.

4 .Continuous random variable X has the following probability density function:
f(x)={ 1/8 0<x<8;
0 otherwise
Continuous random variable Y is defined by Y=ln X .show that P(Y<y)=1/8 (e^y -1)

I'm glad we've got that clarified! If Y= lnX then X= e^Y. In particular, if Y< y then (since exponential is an increasing function) X< e^y. You know, I presume, that with probability density as given,
$$P(X< x)= \int_0^x \frac{1}{8}dt$$
as long as x< 8, 1, if x>= 8.

What is P(X< e^y)?

 

[ngkamsengpeter]

The slope of the graph of the equation y= mx+ b is m and that is equal to the tangent of the angle the line makes with the x-axis: $tan(\theta)= m$. What is the slope for your line and what is the angle?
Also
$$tan(\theta+ \phi)= \frac{tan(\theta)+ tan(\phi)}{1- tan(\theta)tan(\phi)}$$
In particular, tan(45 degrees)= 1 so
$$tan(\theta+ 45)= \frac{1+ tan(\theta)}{1- tan(\theta)}$$
while
$$tan(\theta- 45)= \frac{-1+ tan(\theta)}{1+ tan(\theta)}$$


For the geometric series a+ ar+ ar²+ ...+ arⁿ the sum to the n term is
$$S= a\frac{1- r^{n+1}}{1- r}$$
and the infinite sum is
$$I= a\frac{1}{1- r}$$
How complicated could that be? What are a and r for your example and what is I?


I'm glad to hear that you can do the first part, finding the probability that there are no accidents in a given year- I'll call that p. To find the probability that there are no accident in 2 of 5 given years, use the binomial distribution with probabilities p and 1- p.


I'm glad we've got that clarified! If Y= lnX then X= e^Y. In particular, if Y< y then (since exponential is an increasing function) X< e^y. You know, I presume, that with probability density as given,
$$P(X< x)= \int_0^x \frac{1}{8}dt$$
as long as x< 8, 1, if x>= 8.

What is P(X< e^y)?

For question 2 , due to its modulus , it should become -(I-s)<0.0001<I-s
and this is the complexity I am facing .

For question 3 . If the probability for first section is P(A),then the answer for second section is ( P(A) )^2 X (1 - P(A) )^3 X 5C2 ,is it right ?

For question 4 , do you mean that we can find the answer by finding the answer for P(X<e^y) ? We integrate from 0 to e^y right ?

 

[ngkamsengpeter]

why no reply yet ?

 

[HallsofIvy]

For question 2 , due to its modulus , it should become -(I-s)<0.0001<I-s
and this is the complexity I am facing .[\quote]
Once again, what is I? After you know that, just do the arithmetic- calculate so for several different values of n.

For question 3 . If the probability for first section is P(A),then the answer for second section is ( P(A) )^2 X (1 - P(A) )^3 X 5C2 ,is it right ?

Yes. You said you had already calculated P(A).

For question 4 , do you mean that we can find the answer by finding the answer for P(X<e^y) ? We integrate from 0 to e^y right ?

Yex, if X= ln Y, then Y= e^x.