Question 1155063
<pre>

{{{matrix(1,3,"f(x)",""="",
system(matrix(2,3,
(x-1)/8,if, 1<x<5,
   0   ,"",otherwise)))}}} 

Its graph is this:

{{{drawing(400,280/3,-.5,5.5,-.5,.9, circle(1,0,.07),circle(5.01,.51,.07),
graph(400,280/3,-.5,5.5,-.5,.9),
red(line(1,0,5,.5)),locate(.7,.28,"(1,0)"),
locate(4.5,.8,"(5,0.5)") )}}}

In order for it to be a probability density function, it must be such that when
we shade the area between it and the x-axis, that area must be exactly 1.

{{{drawing(400,280/3,-.5,5.5,-.5,.9,circle(1,0,.07),circle(5.01,.51,.07),
graph(400,280/3,-.5,5.5,-.5,.9), locate(.7,.28,"(1,0)"),
red(line(1,0,5,.5)),
locate(4.5,.8,"(5,0.5)"),

line(1,0,1,0),line(1.028,0,1.028,0.0035),line(1.056,0,1.056,0.007),line(1.056,0,1.056,0.007),
line(1.084,0,1.084,0.0105),line(1.112,0,1.112,0.014),line(1.14,0,1.14,0.0175),line(1.14,0,1.14,0.0175),
line(1.168,0,1.168,0.021),line(1.196,0,1.196,0.0245),line(1.224,0,1.224,0.028),line(1.224,0,1.224,0.028),
line(1.252,0,1.252,0.0315),line(1.28,0,1.28,0.035),line(1.308,0,1.308,0.0385),line(1.308,0,1.308,0.0385),
line(1.336,0,1.336,0.042),line(1.364,0,1.364,0.0455),line(1.392,0,1.392,0.049),line(1.392,0,1.392,0.049),
line(1.42,0,1.42,0.0525),line(1.448,0,1.448,0.056),line(1.476,0,1.476,0.0595),line(1.476,0,1.476,0.0595),
line(1.504,0,1.504,0.063),line(1.532,0,1.532,0.0665),line(1.56,0,1.56,0.07),line(1.56,0,1.56,0.07),
line(1.588,0,1.588,0.0735),line(1.616,0,1.616,0.077),line(1.644,0,1.644,0.0805),line(1.644,0,1.644,0.0805),
line(1.672,0,1.672,0.084),line(1.7,0,1.7,0.0875),line(1.728,0,1.728,0.091),line(1.728,0,1.728,0.091),
line(1.756,0,1.756,0.0945),line(1.784,0,1.784,0.098),line(1.812,0,1.812,0.1015),line(1.812,0,1.812,0.1015),
line(1.84,0,1.84,0.105),line(1.868,0,1.868,0.1085),line(1.896,0,1.896,0.112),line(1.896,0,1.896,0.112),
line(1.924,0,1.924,0.1155),line(1.952,0,1.952,0.119),line(1.98,0,1.98,0.1225),line(1.98,0,1.98,0.1225),
line(2.008,0,2.008,0.126),line(2.036,0,2.036,0.1295),line(2.064,0,2.064,0.133),line(2.064,0,2.064,0.133),
line(2.092,0,2.092,0.1365),line(2.12,0,2.12,0.14),line(2.148,0,2.148,0.1435),line(2.148,0,2.148,0.1435),
line(2.176,0,2.176,0.147),line(2.204,0,2.204,0.1505),line(2.232,0,2.232,0.154),line(2.232,0,2.232,0.154),
line(2.26,0,2.26,0.1575),line(2.288,0,2.288,0.161),line(2.316,0,2.316,0.1645),line(2.316,0,2.316,0.1645),
line(2.344,0,2.344,0.168),line(2.372,0,2.372,0.1715),line(2.4,0,2.4,0.175),line(2.4,0,2.4,0.175),
line(2.428,0,2.428,0.1785),line(2.456,0,2.456,0.182),line(2.484,0,2.484,0.1855),line(2.484,0,2.484,0.1855),
line(2.512,0,2.512,0.189),line(2.54,0,2.54,0.1925),line(2.568,0,2.568,0.196),line(2.568,0,2.568,0.196),
line(2.596,0,2.596,0.1995),line(2.624,0,2.624,0.203),line(2.652,0,2.652,0.2065),line(2.652,0,2.652,0.2065),
line(2.68,0,2.68,0.21),line(2.708,0,2.708,0.2135),line(2.736,0,2.736,0.217),line(2.736,0,2.736,0.217),
line(2.764,0,2.764,0.2205),line(2.792,0,2.792,0.224),line(2.82,0,2.82,0.2275),line(2.82,0,2.82,0.2275),
line(2.848,0,2.848,0.231),line(2.876,0,2.876,0.2345),line(2.904,0,2.904,0.238),line(2.904,0,2.904,0.238),
line(2.932,0,2.932,0.2415),line(2.96,0,2.96,0.245),line(2.988,0,2.988,0.2485),line(2.988,0,2.988,0.2485),
line(3.016,0,3.016,0.252),line(3.044,0,3.044,0.2555),line(3.072,0,3.072,0.259),line(3.072,0,3.072,0.259),
line(3.1,0,3.1,0.2625),line(3.128,0,3.128,0.266),line(3.156,0,3.156,0.2695),line(3.156,0,3.156,0.2695),
line(3.184,0,3.184,0.273),line(3.212,0,3.212,0.2765),line(3.24,0,3.24,0.28),line(3.24,0,3.24,0.28),
line(3.268,0,3.268,0.2835),line(3.296,0,3.296,0.287),line(3.324,0,3.324,0.2905),line(3.324,0,3.324,0.2905),
line(3.352,0,3.352,0.294),line(3.38,0,3.38,0.2975),line(3.408,0,3.408,0.301),line(3.408,0,3.408,0.301),
line(3.436,0,3.436,0.3045),line(3.464,0,3.464,0.308),line(3.492,0,3.492,0.3115),line(3.492,0,3.492,0.3115),
line(3.52,0,3.52,0.315),line(3.548,0,3.548,0.3185),line(3.576,0,3.576,0.322),line(3.576,0,3.576,0.322),
line(3.604,0,3.604,0.3255),line(3.632,0,3.632,0.329),line(3.66,0,3.66,0.3325),line(3.66,0,3.66,0.3325),
line(3.688,0,3.688,0.336),line(3.716,0,3.716,0.3395),line(3.744,0,3.744,0.343),line(3.744,0,3.744,0.343),
line(3.772,0,3.772,0.3465),line(3.8,0,3.8,0.35),line(3.828,0,3.828,0.3535),line(3.828,0,3.828,0.3535),
line(3.856,0,3.856,0.357),line(3.884,0,3.884,0.3605),line(3.912,0,3.912,0.364),line(3.912,0,3.912,0.364),
line(3.94,0,3.94,0.3675),line(3.968,0,3.968,0.371),line(3.996,0,3.996,0.3745),line(3.996,0,3.996,0.3745),
line(4.024,0,4.024,0.378),line(4.052,0,4.052,0.3815),line(4.08,0,4.08,0.385),line(4.08,0,4.08,0.385),
line(4.108,0,4.108,0.3885),line(4.136,0,4.136,0.392),line(4.164,0,4.164,0.3955),line(4.164,0,4.164,0.3955),
line(4.192,0,4.192,0.399),line(4.22,0,4.22,0.4025),line(4.248,0,4.248,0.406),line(4.248,0,4.248,0.406),
line(4.276,0,4.276,0.4095),line(4.304,0,4.304,0.413),line(4.332,0,4.332,0.4165),line(4.332,0,4.332,0.4165),
line(4.36,0,4.36,0.42),line(4.388,0,4.388,0.4235),line(4.416,0,4.416,0.427),line(4.416,0,4.416,0.427),
line(4.444,0,4.444,0.4305),line(4.472,0,4.472,0.434),line(4.5,0,4.5,0.4375),line(4.5,0,4.5,0.4375),
line(4.528,0,4.528,0.441),line(4.556,0,4.556,0.4445),line(4.584,0,4.584,0.448),line(4.584,0,4.584,0.448),
line(4.612,0,4.612,0.4515),line(4.64,0,4.64,0.455),line(4.668,0,4.668,0.4585),line(4.668,0,4.668,0.4585),
line(4.696,0,4.696,0.462),line(4.724,0,4.724,0.4655),line(4.752,0,4.752,0.469),line(4.752,0,4.752,0.469),
line(4.78,0,4.78,0.4725),line(4.808,0,4.808,0.476),line(4.836,0,4.836,0.4795),line(4.836,0,4.836,0.4795),
line(4.864,0,4.864,0.483),line(4.892,0,4.892,0.4865),line(4.92,0,4.92,0.49),line(4.92,0,4.92,0.49),
line(4.948,0,4.948,0.4935),line(4.976,0,4.976,0.497)
))}}}

It's a probability density function because the area between it and the x-axis
is a triangle whose base is 4 and whose height is 1/2 or 0.5 since when we
substitute x=5 in (𝑥−1)/8 we get 4/8 or 1/2 or 0.5 and the area of a triangle is

{{{A=expr(1/2)base*height=expr(1/2)(4)(1/2) = 1}}}

Since the entire area between the graph and the x-axis is 1, it is indeed a
probability density function.  So the problem was telling the truth when it said
this is a probability density function.

The probability that 𝑋 is between two values is the area between the graph and
the x-axis between those two values.
</pre>A. Find the probability that 𝑋 lies between 2 and 4.<pre>
This asks for the area between 2 and 4, which is this shaded area:

{{{drawing(400,280/3,-.5,5.5,-.5,.9,
graph(400,280/3,-.5,5.5,-.5,.9), red(line(1,0,5,.5)), circle(1,0,.07),circle(5.01,.51,.07),

locate(3.5,.7,"(4,0.25)"), locate(1.5,.4,"(2,0.125)"),

line(2,0,2,0.125),line(2.028,0,2.028,0.1285),line(2.056,0,2.056,0.132),line(2.056,0,2.056,0.132),
line(2.084,0,2.084,0.1355),line(2.112,0,2.112,0.139),line(2.14,0,2.14,0.1425),line(2.14,0,2.14,0.1425),
line(2.168,0,2.168,0.146),line(2.196,0,2.196,0.1495),line(2.224,0,2.224,0.153),line(2.224,0,2.224,0.153),
line(2.252,0,2.252,0.1565),line(2.28,0,2.28,0.16),line(2.308,0,2.308,0.1635),line(2.308,0,2.308,0.1635),
line(2.336,0,2.336,0.167),line(2.364,0,2.364,0.1705),line(2.392,0,2.392,0.174),line(2.392,0,2.392,0.174),
line(2.42,0,2.42,0.1775),line(2.448,0,2.448,0.181),line(2.476,0,2.476,0.1845),line(2.476,0,2.476,0.1845),
line(2.504,0,2.504,0.188),line(2.532,0,2.532,0.1915),line(2.56,0,2.56,0.195),line(2.56,0,2.56,0.195),
line(2.588,0,2.588,0.1985),line(2.616,0,2.616,0.202),line(2.644,0,2.644,0.2055),line(2.644,0,2.644,0.2055),
line(2.672,0,2.672,0.209),line(2.7,0,2.7,0.2125),line(2.728,0,2.728,0.216),line(2.728,0,2.728,0.216),
line(2.756,0,2.756,0.2195),line(2.784,0,2.784,0.223),line(2.812,0,2.812,0.2265),line(2.812,0,2.812,0.2265),
line(2.84,0,2.84,0.23),line(2.868,0,2.868,0.2335),line(2.896,0,2.896,0.237),line(2.896,0,2.896,0.237),
line(2.924,0,2.924,0.2405),line(2.952,0,2.952,0.244),line(2.98,0,2.98,0.2475),line(2.98,0,2.98,0.2475),
line(3.008,0,3.008,0.251),line(3.036,0,3.036,0.2545),line(3.064,0,3.064,0.258),line(3.064,0,3.064,0.258),
line(3.092,0,3.092,0.2615),line(3.12,0,3.12,0.265),line(3.148,0,3.148,0.2685),line(3.148,0,3.148,0.2685),
line(3.176,0,3.176,0.272),line(3.204,0,3.204,0.2755),line(3.232,0,3.232,0.279),line(3.232,0,3.232,0.279),
line(3.26,0,3.26,0.2825),line(3.288,0,3.288,0.286),line(3.316,0,3.316,0.2895),line(3.316,0,3.316,0.2895),
line(3.344,0,3.344,0.293),line(3.372,0,3.372,0.2965),line(3.4,0,3.4,0.3),line(3.4,0,3.4,0.3),
line(3.428,0,3.428,0.3035),line(3.456,0,3.456,0.307),line(3.484,0,3.484,0.3105),line(3.484,0,3.484,0.3105),
line(3.512,0,3.512,0.314),line(3.54,0,3.54,0.3175),line(3.568,0,3.568,0.321),line(3.568,0,3.568,0.321),
line(3.596,0,3.596,0.3245),line(3.624,0,3.624,0.328),line(3.652,0,3.652,0.3315),line(3.652,0,3.652,0.3315),
line(3.68,0,3.68,0.335),line(3.708,0,3.708,0.3385),line(3.736,0,3.736,0.342),line(3.736,0,3.736,0.342),
line(3.764,0,3.764,0.3455),line(3.792,0,3.792,0.349),line(3.82,0,3.82,0.3525),line(3.82,0,3.82,0.3525),
line(3.848,0,3.848,0.356),line(3.876,0,3.876,0.3595),line(3.904,0,3.904,0.363),line(3.904,0,3.904,0.363),
line(3.932,0,3.932,0.3665),line(3.96,0,3.96,0.37),line(3.988,0,3.988,0.3735),line(3.988,0,3.988,0.3735)   )}}}

This is a trapezoid (or 'trapezium' if you live in the UK), and if you turn your
head 90° it has two bases b<sub>1</sub>=0.125 and b<sub>2</sub>=0.25, and 
height = 2.  The formula for the area is
{{{A=expr(1/2)(b[1]+b[2])h=(0.5)(0.125+0.375)(2) = 0.5=1/2}}} 
So that's  the answer.
</pre>B. Find the probability that 𝑋 is less than 3.<pre>
{{{drawing(400,280/3,-.5,5.5,-.5,.9,
graph(400,280/3,-.5,5.5,-.5,.9), locate(.7,.28,"(1,0)"), circle(1,0,.07),circle(5.01,.51,.07),
red(line(1,0,5,.5)),
locate(2.5,.5,"(3,0.25)"),

line(1,0,1,0),line(1.028,0,1.028,0.0035),line(1.056,0,1.056,0.007),line(1.056,0,1.056,0.007),
line(1.084,0,1.084,0.0105),line(1.112,0,1.112,0.014),line(1.14,0,1.14,0.0175),line(1.14,0,1.14,0.0175),
line(1.168,0,1.168,0.021),line(1.196,0,1.196,0.0245),line(1.224,0,1.224,0.028),line(1.224,0,1.224,0.028),
line(1.252,0,1.252,0.0315),line(1.28,0,1.28,0.035),line(1.308,0,1.308,0.0385),line(1.308,0,1.308,0.0385),
line(1.336,0,1.336,0.042),line(1.364,0,1.364,0.0455),line(1.392,0,1.392,0.049),line(1.392,0,1.392,0.049),
line(1.42,0,1.42,0.0525),line(1.448,0,1.448,0.056),line(1.476,0,1.476,0.0595),line(1.476,0,1.476,0.0595),
line(1.504,0,1.504,0.063),line(1.532,0,1.532,0.0665),line(1.56,0,1.56,0.07),line(1.56,0,1.56,0.07),
line(1.588,0,1.588,0.0735),line(1.616,0,1.616,0.077),line(1.644,0,1.644,0.0805),line(1.644,0,1.644,0.0805),
line(1.672,0,1.672,0.084),line(1.7,0,1.7,0.0875),line(1.728,0,1.728,0.091),line(1.728,0,1.728,0.091),
line(1.756,0,1.756,0.0945),line(1.784,0,1.784,0.098),line(1.812,0,1.812,0.1015),line(1.812,0,1.812,0.1015),
line(1.84,0,1.84,0.105),line(1.868,0,1.868,0.1085),line(1.896,0,1.896,0.112),line(1.896,0,1.896,0.112),
line(1.924,0,1.924,0.1155),line(1.952,0,1.952,0.119),line(1.98,0,1.98,0.1225),line(1.98,0,1.98,0.1225),
line(2.008,0,2.008,0.126),line(2.036,0,2.036,0.1295),line(2.064,0,2.064,0.133),line(2.064,0,2.064,0.133),
line(2.092,0,2.092,0.1365),line(2.12,0,2.12,0.14),line(2.148,0,2.148,0.1435),line(2.148,0,2.148,0.1435),
line(2.176,0,2.176,0.147),line(2.204,0,2.204,0.1505),line(2.232,0,2.232,0.154),line(2.232,0,2.232,0.154),
line(2.26,0,2.26,0.1575),line(2.288,0,2.288,0.161),line(2.316,0,2.316,0.1645),line(2.316,0,2.316,0.1645),
line(2.344,0,2.344,0.168),line(2.372,0,2.372,0.1715),line(2.4,0,2.4,0.175),line(2.4,0,2.4,0.175),
line(2.428,0,2.428,0.1785),line(2.456,0,2.456,0.182),line(2.484,0,2.484,0.1855),line(2.484,0,2.484,0.1855),
line(2.512,0,2.512,0.189),line(2.54,0,2.54,0.1925),line(2.568,0,2.568,0.196),line(2.568,0,2.568,0.196),
line(2.596,0,2.596,0.1995),line(2.624,0,2.624,0.203),line(2.652,0,2.652,0.2065),line(2.652,0,2.652,0.2065),
line(2.68,0,2.68,0.21),line(2.708,0,2.708,0.2135),line(2.736,0,2.736,0.217),line(2.736,0,2.736,0.217),
line(2.764,0,2.764,0.2205),line(2.792,0,2.792,0.224),line(2.82,0,2.82,0.2275),line(2.82,0,2.82,0.2275),
line(2.848,0,2.848,0.231),line(2.876,0,2.876,0.2345),line(2.904,0,2.904,0.238),line(2.904,0,2.904,0.238),
line(2.932,0,2.932,0.2415),line(2.96,0,2.96,0.245),line(2.988,0,2.988,0.2485),line(2.988,0,2.988,0.2485),
line(3.016,0,3.016,0.252)
))}}}

the shaded area between between the graph the x-axis is a triangle whose base is
2 and whose height is 1/4 or 0.25 since when we substitute x=2 in (𝑥−1)/8 we get
1/8 or 0.125 and the area of a triangle is 

{{{A=expr(1/2)base*height=expr(1/2)(2)(1/4) = 1/4 = 0.25}}}

That's the answer.

Edwin</pre>