If the diameter X of axles has the density fIx} = k if 119.9 :=; x ~ 120.1 and 0

Graph the probability function \(f(x) = kx^{2}\) (x = 1, 2, 3,4, 5; k suitable) and the distribution function. Text Transcription: f(x) = kx^2

Graph the density function \(f(x)=k x^{2}(0 \leqq x \leqq 5\): k suitable) and the distribution function. Text Transcription: f(x) = kx^2 (0 <= x <= 5

(Uniform distribution) Graph f and F when the density is f(x) = k = const if \(-4 \leqq x \leqq 4\) and 0 elsewhere. Text Transcription: -4 <= x <= 4

In Prob. 3 find \(P(0 \leqq x \leqq 4)\) c such that P( -c < X < c) = 95%. Text Transcription: P(0 <= x <= 4)

Graph f and F when f{ -2) = f(2) = 1/8, f( -1) = f(1) = 3/8. Can f have further positive values?

Graph the distribution function \(F(x) = I - e^{-3x}\) if x > 0, F(x) = 0 if \(x \leqq 0\), and the density f(x). Find x such that F(x) = 0.9. Text Transcription: F(x) = I - e^-3x x <= 0

Let X be the number of years before a particular type of machine will need replacement. Assume that X has the probability function f(l) = 0.1, f(2) = 0.2, f(3) = 0.2, 1(4) = 0.2. 1(5) = 0.3. Graph I and F. Find the probability that the machine needs no replacement during the first 3 years.

If X has the probability function \(f(x) = k/2^{x}\) (x = O. l. 2 ... '). what are k and \(P(X \geqq 4)\)? Text Transcription: f(x) = k/2^x P(X >= 4)

Find the probability that none of the three bulbs in a traffic signal must be replaced during the first 1200 hours of operation if the probability that a bulb must be replaced is a random variable X with density \(f(x) = 6[0.25 - (x - 1.5)^{2}]\) when \(1 \leqq x \leqq 2\) 2 and f(x) = 0 otherwise. where x is time mea~ured in multiples of 1000 hours. Text Transcription: f(x) = 6[0.25 - (x - 1.5)^2] 1 <= x <= 2

Suppose that certain bolts have length L = 200 + X mm, where X is a random variable with density \(f(x)=\frac{3}{4}\left(1-x^{2}\right)\) if \(-1 \leqq x \leqq 1\) and 0 otherwise. Determine c so that with a probability of 95% a bolt will have any length between 200 - c and 200 + c. Hint: See also Example 5. Text Transcription: f(x) = 3/4 (1 - x^2) -1 <= x <= 1

Let X [millimeters] be the thickne~~ of washers a machine turns out. Assume that X has the density f(x) = h if 1.9 < x < 2.1 and 0 otherwise. Find k. What is the probability that a washer will have thickness between 1.95 mm and 2.05 mm?

Suppose that in an automatic process of filling oil into cans, the content of a can (in gallons) is Y = 50 + X. where X is a random variable with density fIx) = I - Ixl when Ixl :=; 1 and 0 when Ixl > I. Graph fIx) and F(x). In a lot of 100 cans, about how many will contain 50 gallons or more'? What is the probability that a can will contain less than 49.5 gallons? Less than 49 gallons?

Let the random variable X with density \(f(x) = ke^{-x}\) if \(0 \leqq x \leqq 2\) and 0 otherwise (x = time measured in years) be the time after which cel1ain ball bearings are worn out. Find k and the probability that a bearing will last at least I year. Text Transcription: f(x) = ke^-x 0 <= x <= 2

Let X be the ratio of sales to profits of some fiml. Assume that X has the distribution function F(x) = 0 if x < 2, \(F(x) = (x^{2} - 4)/5\) if \(2 \leqq x<3\) F(x) = I if \(x \geqq 3\). Find and graph the density. What is the probability that X is between 2.5 (40% profit) and 5 (20% profit)? Text Transcription: F(x) = (x2 - 4)/5 2 <= x < 3 x >= 3

Show that b < c implies \(P(X \leqq b) \leqq P(X \leqq c)\). Text Transcription: P( X <= b) <= P(X <= c)

If the diameter X of axles has the density fIx} = k if 119.9 \(\leqq x \leqq 120.1\) and 0 otherwise, how many defectives will a lot of 500 axles approximately contain if defectives are axles slimmer than 119.92 or thicker than 120.08? Text Transcription: <= x <= 120

Let X be a random variable that can as~ume evelY real value. What are the complements of the events \(x \leqq b, X<b . X \geqq c, X>c, b \leqq X \leqq c, b<X \leqq c\) Text Transcription: X <= b, X < b, X >= c, X > c, b <= X <= c, b < X <= c

A box contains 4 right-handed and 6 left-handed screws. Two screws are drawn at random without replacement. Let X be the number of left-handed screws drawn. Find the probabilities P(X = m. P(X = I). P(X = 2). P( I < X < 2), \(P(X \leqq 1), P(X \geqq 1)\), P(X > I), and P(0.5 < X < 10). Text Transcription: P(X <= 1), P(X >= 1)