MATH 3C Lecture Notes - Lecture 6: Standard Deviation, Cumulative Distribution Function, Antiarrhythmic Agent

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violetmanatee539

9 May 2016

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1: suppose the ra(cid:374)do(cid:373) (cid:448)aria(cid:271)le x is the dista(cid:374)(cid:272)e (cid:894)i(cid:374) kilo(cid:373)eters(cid:895) fro(cid:373) a gi(cid:448)e(cid:374) poi(cid:374)t to the (cid:374)earest (cid:271)ird"s nest with the probability density function given by (cid:4666) (cid:4667) for x(cid:1096)0 a(cid:374)d f(cid:894)x(cid:895)= 0 other(cid:449)ise. Prove that f(x) is a probability density function. Find the probability that there is a bird nest within kilometer of the given point. What is the average wait time: a swarm of bees is released from a certain point. The proportion of the swarm located at least 2 meters from the point of release after 1 hour is a random variable that is exponentially distributed with parameter c = 2. Find the average proportion under the given condition. The elimination rate for this drug is . 4 per hour. This means d[y(t)]/dt = -. 4 y(t) where y(t) represents the amount of drug in the body at time t. y is measured in milligrams and t is measured in hours.

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Show transcribed image text This Question: 1 pt 12 of 44 (0 complete) This Quiz: 44 pts possible Verify Property 2 of the definition of a probability density function over the given interval. ffx) 22x What is Property 2 of the definition of a probability density function? O A. The area under the graph of f over the interval [a,bl is b O B. The area under the graph of f over the interval [a.b] is 1 O C. The area under the graph of f over the interval [a,b is a. Since the given interval. [0.oo), has a right-endpoint of infinity, the area under the graph of fwill be calculated using an improper integral. Choose the correct formula for an improper integral over the interval [a,ool below t(x) dx= lim | f(x) dx= lim [F(x)]a= lim (F(b)-F(a)) b-o0 f(x) dx = lim | f(x) dx = lim [F(x)]-lim (F(a)-F(b)) b- 00 b--0o f(x) dk- lim x) dx- lim [F(x)m (F(b)-F(a)) b-+00
O D. 00 f(x) dx= lim f(x) dx= lim [F(æ£´= lim (F(a)-F(b)) b--oã Substitute a and b into the left side of the formula from the previous step and express the integral as a limit. area 2edx lim2edx Next, determine F(x). First, find the antiderivative of f. 2(Use C as the artbitrary constant.)
Next, determine F(x). First, find the antiderivative of f. as the anbliay cosat) Let C 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over [0,oo) using the far right side of the formula for the area. area = As b approaches oo, e approaches infinity where c is a constant. Use this fact to calculate the limits from the previous step area+1 Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below A Property 2 o the definition o a probability density unction over the given interval has been ver ed since the expression in the previous step simplifies to o B. Property 2 of the definition of a probability density function over the given interval has not been verified because the expression in the previous step does not simplify to the expected area value. Â° C. â D Property 2 ofthe definition of a probability density function over the given interval has been verified since the expression in the previous step simplifies to 0 Property 2 o the definition o a probability density unction over the given interval has been ve ned since the expressio n e evious ste si plifies to 1.

violetjellyfish535

Probability theory tells us that if the average time between events is k, the probability that x, the time between events, is between a and b is given by,

P(a ≤ x ≤ b) = ∫ ^b_af(x) dx ------ equation (1)

f(x) = 0, x < 0 ------------- equation (2)

= ke^-kx , x ≥ 0 ------------ equation (2)

Suppose that at a busy intersection of a city, traffic accidents occur at an average rate of one every three months. After residents complained, changes were made to the traffic lights at the intersection. It has now been eight months since the changes were made and there have been no accidents.

1(a) Find f(x) for accidents are occurring at a rate one of every 3 months.

[Hint: use f(x) from equation (2)]

1(b) Were the changes effective or is the 8-month interval without an accident a result of chance?

[Hint: Compute P(x ≥ 8) by using f(x) from (a) and decide on the obtained

result.]

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MATH 3C Lecture Notes - Lecture 6: Standard Deviation, Cumulative Distribution Function, Antiarrhythmic Agent

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